Btw, is there a problem by sending BTC to a centralized exchange through Tor? in terms of getting the transaction blacklisted or something? I just use Tor by default for common sense privacy, but you never know nowadays.
With Bitcoin Core you are sending Bitcoin through its blockchain. Or even better said, you don't send bitcoin, you just create and sign a transaction which is picked up by the other nodes.
And the other nodes don't care neither about Tor, nor about the fact the recipient address is in the wallet of a CEX.
Does anyone know if BTC Core will work properly if installed on Tails USB with Persistent Storage enabled?
I can't help with that, sorry. My Linux knowledge is limited and I didn't manage to install and use all the software I wanted to a Tails USB.