Personally, I would just sign up as a batch provider with EFTPS, get a "Reporting Agent Authorization" (form 8655) from the client, and make the deposit for them. After that, you would be prepared to handle the issue if it ever came up again. It is free and simple to do.
I have never had a broker who was actually allowed to do a tax deposit if it wasn't done through their corporate channels. That could simply be because of the brokers I work with, but I have always been told that they are not allowed to assume that liability.
As for deposit coupons, can you even do that anymore? I thought they were phased out after the electronic deposit requirement and after a certain point they would no longer accept coupons...