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.