Create an account just for T-mobile. With someone like Fidelity setting up a new account is a few clicks. Ally makes it pretty easy too. Then set it up to fund automatically from another account within a certain limit. Or if your monthly bill is stable, setup a schedule to fund it monthly.

What are the rights of the T-mobile customer in the USA, if they somehow do mess up an automatic bill payment?

I'm searching using the European/international term for this sort of payment ("direct debit usa") which gives me API documentation from many American payment providers, but nothing about the customer's rights. I assume it's the technical term there, and there's another term used with the public.

(In Britain and the EU this kind of bill-paying agreement has very strong rights for the payer, they can ask their bank to reverse any transaction for a long time without giving a reason. My online banking interfaces have easy buttons to do this, or to block future transfers.)

You can easily create an account just for T-Mobile and auto-deposit into it monthly.