What is the best test with the least radiation to find out if the bone(s) near #1 upper tooth or plate cracked?

Bone: It's very rare to see a fracture around that area, unless you had an extraction or trauma.

Hmmmm.: Unfortunately, potential cracking of the alveolar bone, and even the removal of alveolar bone from around a tooth is not uncommon. This is one of the possiblitiies of any surgery that involves removing a tooth. I would certainly advise to have the area checked by the oral surgeon that did the surgery. Usually, this breakage of the bone wiill eventually heal.

Cone Beam CT: A digital cone beam ct delivers a 3d image of your jaw bones with very little radiation in comparison to a hospital grade ct machine. It spins around your head for about 10 seconds and gives an extrextremely accurate 3d rendering.