I would not take a chance even if it was/is legal. All you need to happen is to get pulled out of line and get accused of being a money laundering kingpin or just having it seized under some BS premis. Are you going to fight to get your money back on foreign soil? Good luck with that. But that's just me.
I had all that and have been continually refused due to having a METV. I am trying to avoid having to get insurance and a police background report to file for my retirement visa in the US. It's going to be cheaper and easier to use an agent.
Immigration in Phuket and I am sure others will give you a letter if you have a TM30 copy from a hotel if you are staying one day. That solves that issue fast.
It should be with the right visa. Tourist visa is a dead stop but otherwise any other visa and residence certificate and passport should work. I have been to over 20 branches of banks in Bangkok and Phuket. Most told me I would be OK with any other visa other that tourist visa.
If you are on a tourist visa your only real hope is an agent. If you have any other visa 90 days or longer you should be able to get a Bangkok Bank account providing you have your passport with visa and residence form from immigration.