The externality applies to residents who do not take public transit too. The two decisions should not be coupled.
If you want to do a Pigovian tax, do a Pigovian tax properly. If you want to do congestion pricing, make it budget neutral to users. If you want transfer payments, do that and call it that.
but it's purpose is to lower congestion, and the fee varies with time of day based on congestion, and the people who pay it are contributing to the congestion so ... it's a congestion priced fee. Not clear it's a tax at all, in the same way a fishing permit is not called a tax.
The money isn't going to residents.