Yeah, it may be installed, Over there is the guide on how to do this, but take into account that there will be a need for at least 1 TB flash stick to accommodate the full blockchain ( as 512 GB will be not enough for this however it may be less if you run Bitcoin Core in pruned mode) Also you will need HW to sign transactions when Core is in watch-only mode.
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.
Commonly, there should be no problem, anyway you can always check with the small sum.