I propose we abandon the notion of even having a capital city. Instead of a capital, each branch has its own arrangements:

Legislative branch: Congress is supposed to be the branch that most closely represents the people. The site where Congress meets relocates after each census to the mean center of US population. They'll be prohibited from constructing buildings of any kind. Congressional meetings will occur in rented spaces like high school gyms, convention centers, etc. Congress members won't have to worry about expensive apartments, as they'll be expected to sleep in quarters supplied to them, often in old dorm rooms, former homeless shelters, and other such communal living facilities.

The Supreme Court: They're supposed to be above it all. Let's make that literal. They will be relocated to a new building constructed on the peak of Mt. Denali, the highest mountain in the US. SCOTUS justices once confirmed to the court will be expected to relocate to their new home. They will live in simple on-site monastic cells. The only way they are ever allowed down from the mountain is if they resign their seat or die of old age. All petitions to the court must be delivered on parchment scrolls carried up the mountain by hand. The only communication they are allowed with the outside world are court briefs and rulings.

The executive: the president gets to literally be homeless. To keep them humble, the president will be ineligible to own private property of any kind. They will either have to sleep on the sidewalk or travel from home to home, accepting shelter from whomever will give it to them. They can stay on the property of any one person no more than 3 days over their entire presidential term. Also we'll make it perfectly legal to kill the current sitting president. The president literally does not get the protection of the law. Let the most powerful be the most vulnerable.