You can't get persistent malware if all browser history and installed apps disappear every time you reboot. Yes, TAILS was designed for security. Yes, TAILS was also designed for privacy. Obviously its no longer anonymous if you log into your bank.
Tor Browser is by far the most hardened & secure web browser. If a bank is blocking Tor, that bank probably doesn't understand security and its a red flag. Choose another bank. Or, better, use monero. Its way more secure than banks.
Disclaimer: I used to work in info security for a bank in Europe.