If you found a branch who is willing to talk to you, you're good.
From my experience I needed a certificate of residence, my visa document and my passport to open an account. Not sure if a Thai phone number was required but they took mine.
Note; every bank and every branch seem to do their own thing. A couple of banks turned me away cold and others were happy to help.
You need a residence certificate and a doctor's certificate beforehand. Both easy to obtain. Once you have them, make an appointment, go do some minor eye tests and they issue you with a Thai license. I forgot the cost but it's not much.
You're going to wait a little bit especially if there is a crowd. I think it took me about 2 hours from arriving to walking out with my driver licenses in Rayong.
If you don't mind the risk you are fine. I'd play safe and flying into Cambodia or Laos and then border cross. They won't stick you on a plane back to the UK/US or whatever at a border checkpoint.
The concern I have with issues like this is when the rubber hits the road, you are going to be at the mercy of the immigration officer's understanding of the matter. You may be 100% fine.