I mean, if Satan is portrayed as the Lord of Hell, him punishing people that haven't died yet makes even less sense.

Besides, the cases of people being punished for sin in life (like drowning damn near everyone, smiting Sodom and Gomorrah or striking down Ananias for a stupid lie) I'm aware of are still all God. If anything, the Serpent would be the tempter leading people to sin.

In some places (notably Rev. 12), Satan is the accuser. That would make him part of the sinners' punishment in the afterlife (unless Jesus intercedes on behalf of his faithful). However, I don't think I've seen that framing used much in popular depictions, and it still wouldn't make him a lord of anything.

So I don't know of any biblical source that would make any of the entities typically conflated with the Devil the Lord of Hell and the punisher of sinners