You can use TOR browser at your own risk, no matter how good TOR browser is in giving you another US IP or even a New York one IP the problem with TOR is it makes your computer receive a lot of unusual traffic (making you anonymous if you follow the TOR guidelines) but also maximizes the risk of your exchange account getting hacked very soon, so for me its not a very good idea to use TOR in this situation.
OK what about a VPN? Or is that just a heightened version of the Tor browser on a larger scale ?
With a very legit and trustworthy VPN you have a better security than TOR regarding the possibility of your account getting hacked in the exchange because the VPN assign you only 1 IP so it should be safe. However the authorities can still track you when using VPN but that takes time so I think they will not bother, thats the only downside.