Bus fares make up only like 2% of the income for my cities public transit system. They can afford to go free.

Sure but why would a fee do that? If anything that would encourage this behavior because if I pay 2$ I feel entitled to spend there as much time as I want.

This is pretty well established theory now in behavior economy:

https://en.m.wikipedia.org/wiki/Motivation_crowding_theory

We already invented the most effective and humane fix for this - social workers. I don't think anything can work better then another person trained in handling social situations correcting the issue themselves.