a - give up your details when required to do so by the government.
b - lie on a government form and/or to a government official
If you choose b, then at any point in the future it's discovered that you did lie, then this can make life very very difficult. Not only can your Visa be cancelled, it can mean you're never again able to visit the US. It means uprooting your entire life to return to the country you were born in.
There's been plenty of stories, some posted here, where this has happened to someone who either through a mistake, or even no fault of their own - has ended up in hot water and their Visa withdrawn.
Using Tor and successfully remaining truly anonymous are non-trivial things, and endangering your livelihood and ability to remain in the US is a large risk.
This ignores the genuine possibility of prosecution for making false statements and/or lying to a government official.
At the point your threat model includes nation states of note — in my opinion, any recommendation of Tor should acknowledge why Tor would not increase threats instead of reducing them.
You should consider using Tor. It was first and foremost designed for this very purpose.