I'm a Tails user. I strongly advise against attempting to download the blockchain onto a USB stick. Relying on such persistent storage for an extended duration is not recommended. In my two years of using Tails, I've encountered USB corruption issues numerous times, and must have reinstalled the OS by more than five times. It's also crucial to upgrade to the latest version promptly, which further complicates using it for this purpose.
Btw, is there a problem by sending BTC to a centralized exchange through Tor? in terms of getting the transaction blacklisted or something?
Very probable. I doubt you'll even reach the stage where it shows your deposit address. Your IP will likely have been blacklisted by then.
Edit: After reading NeuroticFish's response below, I'm uncertain if you're referring solely to broadcasting through Tor while accessing the exchange from the clearnet. I understand that you're accessing the exchange through Tails and consequently via Tor as well.
I would be running the OS through an USB but I would have a portable SSD to host the blockchain and keep it updated there. Or I could just install it on the SSD and use it as the OS there. I would be accessing the CEX with clearnet. So when I send to a CEX, I would like to be able to send through clearnet, so it matches the clearnet IP with the broadcast IP. If I send it through my node, then it would go directly into the CEX node, so they would know the IP, or it goes through additional hops? I've read people got transactions blocked from broadcasting through Tor into a CEX.
Sometimes I want privacy so I want to run through Tor, sometimes I want a clearnet transaction. If I use Tails, is it possible to send through clearnet? or is it all forced through Tor? You can use clearnet browser if you want to, not sure how this would work with a BTC transaction.