...as a non-US resident? Regardless whether I can buy it or not.
If not, what's the best shooting country in the EU and how do I go about acquiring said shotgun?
In the U.S., you cannot buy firearms unless you're an official U.S. citizen. However, it is entirely possible to rent firearms and shoot them at a shooting range that offers rental firearms. People visiting from out of the country tend to be able to do that. Just don't go to states like California, New York, Illinois, Washington, New Jersey, and any of the other heavily blue states. Still, if you intend to use a shotgun to off yourself, that can be pretty hard to do at a shooting range as others will be present, and might be able to quickly intervene and stop you. Depends on timing I suppose, how many people are around, and how quickly you can aim the shotgun in your mouth before someone tackles you.
As far as buying guns in the EU, I'm afraid I'm not much of an expert. I know it is possible to buy shotguns and lever-action rifles in the U.K., and I believe Switzerland also allows for ownership of firearms. Still, as is with anything firearm-related in the EU, you'll probably have to jump through some hoops with getting enough licensing and certification to be able to buy a firearm. Which country are you in?