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.