Never heard of a AAAT but I do know that Americans can get a PAL which is a POSSESION AND AQUISITION license. That would be your best solution if you plan on bringing firearms into Canada on a regular basis. Also, remember the only way to bring in a handgun is if you are going to a sanctioned shoot at a registered handgun club. General travel in Canada other than to the range holding that shoot is illegal and will land you in jail! So best not bother with handguns.