I've got the same situation at one of my banks. I set up an automated scheduled transfer on the first of every month from one of my accounts at that bank to the other, and another transfer back to the first account a few days later.

I get the feeling that eventually they'll see what's going on, and try to charge the fee anyway. If that ever happens, I'll just start using an account at another bank as the source of the first transfer, and another as the destination of the second transfer.

So yes, that service exists. Any bank with halfway decent online banking lets you do it yourself, for free.