Bauman was later beaten to death with a steel pipe by a Tsarist monarchist in the aftermath of the first Russian Revolution of 1905 as both the far right and revolutionary left begun skirmishing in the streets of Moscow. His death made him a martyr that would help build the Moscow Soviet and solidified the Bolshevik's hold in the city. By the standard of Bolshevik ethics, being the radical teleological and historicist worldview (historical materialism) hitched to the party goal of building and succeeding the Russian Revolution, Lenin made the correct decision to retain an excellent tool for achieving the goal of the Party. You have to understand that the Bolsheviks operated on what we westerners consider a completely alien moral system. If we were to resurrect any of the old guard, they'd outright dismiss entertaining the question of "what if bauman was ejected" as menshevik idealism purely because concrete material reality of what occurred is the ultimate judge of correctness. Lets examine deeper into this decision and the attempt to understand the decision process behind it.

Lenin in the period of fall 1902 to the spring of 1903 lived in a quite tumultuous world. The Boxer Rebellion had occurred a few years prior and was squashed by the Imperialist powers that had sliced the wealth of Qing China amongst themselves, Tsarist Russia being one of the players who took the rebellion as an opportunity to move its armies into Manchuria, present day North-East China, and the Empire at this point in time attempting to solidify its hold on Manchuria while slowly trying to worm its way into Korea - the economic domain of the quietly growing Imperial Japan - as well as completing the great trans-siberian railway that would connect the Tsar's lands from the Baltic Sea to the Pacific Ocean (more or less). The Empire itself was in the throes of transforming from a backwards feudalistic economic mode of production to the modern capitalist system with the aid of foreign finance capital and domestic lordlings or venture capitalists. This of course brings the same economic hardships that was seen in the industrial revolution in Marx's time a few generations back in England, France, Germany, and the rest of western Europe. Peasants in the Commons being thrown to the cities to be transformed into proletarians and used as fuel and grease for the factories. Socialists of all flavors flowing amongst the exploited workers and building class consciousness among them. In fact every year since the Boxer Rebellion in 1900 has been a busy year for Lenin and the Bolsheviks both in Exile and in the Field as every year since has had sparks of labor uprisings and strikes both spontaneously or lead by the Bolsheviks in their respective regional cells. Quoting the book on the history of the Party

A May Day strike at the Obukhov munitions plant in St. Petersburg in 1901 resulted in a bloody encounter between the workers and troops. The only weapons the workers could oppose to the armed forces of the tsar were stones and lumps of iron. The stubborn resistance of the workers was broken. This was followed by savage reprisals: about 800 workers were arrested, and many were cast into prison or condemned to penal servitude and exile. But the heroic "Obukhov defence" made a profound impression on the workers of Russia and called forth a wave of sympathy among them.

In March 1902 big strikes and a demonstration of workers took place in Batum, organized by the Batum Social-Democratic Committee. The Batum demonstration stirred up the workers and peasants of Transcaucasia.

In 1902 a big strike broke out in Rostov-on-Don as well. The first to come out were the railwaymen, who were soon joined by the workers of many factories. The strike agitated all the workers. As many as 30,000 would gather at meetings held outside the city limits on several successive days. At these meetings Social-Democratic proclamations were read aloud and speakers addressed the workers. The police and the Cossacks were powerless to disperse these meetings, attended as they were by many thousands. When several workers were killed by the police, a huge procession of working people attended their funeral on the following day. Only by summoning troops from surrounding cities was the tsarist government able to suppress the strike. The struggle of the Rostov workers was led by the Don Committee of the R.S.D.L.P.

As you can gather, Lenin was very busy. The party that existed in Tsarist Russia was stretched by leagues and the agents at Lenin's disposal were too few for the ground that was needed to be covered. Every single cadre that acted as an agent of the central committee was a professional Revolutionary who professed of their own free will to rationally subordinate themselves to the party. To maintaining iron discipline to execute the Party's orders without question, to abandon family, to face death without hesitation. To live an ascetic life without property, stable employment, legal identity as needed, and subsist off of the allowance of the Party as needed in order to freely move where the Party needs them. To internalize and embody Bolshevik partiinost and understand that the Party is an instrument of history - Not a group of men and women with opinions contrary to the status quo but the embodiment of proletarian class consciousness. To oppose the Party is to oppose the working class; to oppose the working class is to oppose history itself. - and that they themself are a cell, filled with scientific consciousness mediated by Marxist theory, within the party and of the party thus must embody and act upon the party line to accomplish the Party's primary goal of the liberation of the working class. To cultivate to their best ability the skillset of Revolutionary Conspiracy, ranging from printing, cryptography, armed combat, agitation, forgery, administration, labor organization, criminal actions, etc. These cadre agents are people that operate miles and countries away from Lenin and the other members of the Central Committee, expected to operate autonomously from the party yet be the very face of the party enacting its very will. These kinds of dedicated people are few and far between, and even having an agent that can embody a fraction of these criterion and be trusted to do the will of the party are very valuable gameboard pieces. We have to assume with the scarce knowledge we currently have on Bauman and his life that over the course of his revolutionary career up to this point he had shown the Central Committee and Lenin skill, competence, results and loyalty to a significant degree that expelling him for his personal indiscretions and later horrific behavior would've costed the party more greatly in its operational programs than it did retaining him and suffering the backlash from the questionable morality of the party for protecting a wretched man as him.

My personal theory on why Lenin did so is as follows with the questions he probably asked and answered himself: Who or what group demanded the expelling of this agent? Which class and which political groups benefitted by the expelling of this agent? What precedent would be set by expelling this agent? Does expelling this agent help build the revolutionary movement to overthrow the Tsar and begin the Communist Revolution?

Later in the summer of this year of 1903, the Second Congress of the Russian Social Democratic Labor Party and the great split between the Bolsheviks and Mensheviks was to occur, and two of the future leading voices of the Mensheviks, Julius Martov and Pavel Axelrod respectively, were the two most outspoken voices calling for the expulsion of Bauman. Without getting too far into the weeds of history, the Bolsheviks and Mensheviks had different visions for what a revolutionary political party should be and for Martov and Axelrod who desired a mass party instead of a vanguard party did not see the same way as Lenin did for the necessity of professional revolutionaries and believed the RSDLP would have a net gain expelling him over retaining him. The Bauman affair was a footnote to the developing ideological differences between the two factions ultimately but did contribute to the split at the congress.

To Lenin, it was obvious that the Tsar and the Capitalists would ultimately benefit from expelling Bauman as it would prove that they could utilize moral outrage to pressure the RSDLP into cutting off ugly portions of itself to conform to their bourgeois morality as well as defang the Party of capable revolutionary agents fomenting revolution across the Empire.

And the Precedent I would guess Lenin saw that would be set was what was previously mentioned that by showing that external moral tribunals could hold power over the party, be that liberal public opinion, bourgeois law, etc. it would fatally compromise the party's autonomy and its ability to make decisions based solely on revolutionary expediency towards its goals.

Ultimately, these lines of thought would lead to the conclusion that expelling Bauman would ultimately be an action that hinders the movement to overthrow the Tsar and launch the Revolution. Which leads me back to my earlier statement that I wouldn't want to be in Lenin's position making such a decision because I recognize I still have significant bourgeois baggage that would hinder me from making any such cold analytical decisions in finding the correct path for the Revolution.