What I don't understand on your statement is on how are you going to enjoy your money if you will not withdraw it to your bank account?
I highly recommend coinbase pro: https://pro.coinbase.com/markets
They have a license to operate and their headquarters is in San Francisco California so quite safe to trade fiat to BTC. I believe bitfinex offer BTC/USD trading too but keep away on that exchange since they have a lawsuit against them.
You can check your state on the below link if coinbase has a licensed to operate in your place.
See here: https://www.coinbase.com/legal/licenses
FYI, you already answered your question. Just use crypto and you don't need a bank account. What's the point?
Let me clarify my question, I want to trade my coins for USD and vice-versa but I don't plan to withdraw my USD to bank.
For example, say the price of bitcoin reaches $10k so I would sell some and keep the USD in the exchange, and say when the dip happens and the price goes to $8k I plan to buy back the coins using USD stored on exchange and withdraw them back to my wallet.