Other than the big negative of having to deal with traveling to the US there is nothing illegal about non-citizens handling weapons while they are here. As I understand it there is a fair amount of firearm tourism that happens on the west coast from Japan.
Your biggest problem is likely to be finding loaner firearms. It is worth inquiring with the people offering the training to find out if there is any way to use borrowed firearms. Usually purchasing ammunition while here isn’t a big deal but .22LR is hard to find today.
I believe that it is also mostly legal to bring firearms into the country for training or sporting purposes. You would have to be careful though since the largest international airports are in states with very restricted firearm laws.
Oh, and make sure you don’t fire any weapons for at least a day before you try to leave the country. The TSA sniffers will catch powder residue from shooting firearms if you have to go through one.