Lighting, Light Fixtures, Ceiling and Exhaust Fans - replacing light with on that must be "supported by the building structure"

I am attempting to replace two existing bathroom vanity lights with heavier ones. The new fixtures instructions specifically say: "These fixtures are intended to be mounted to and outlet box. The box must be directly supported by the building structure." The new fixtures have round mounting plates to attach to the outlet box.

Because the old fixtures were light weight, they were attached to the wall with simple drywall anchors. There are no outlet boxes, just romex coming out of a hole in the drywall. One of the old fixtures wiring has a stud right next to it, so I can add a box to that stud. The other fixture needs to be mounted between two studs. So here is my dilemma:

How do I add an outlet box that will support the weight of the new fixture without causing any more damage to the drywall? Is there a ceiling fan type mount that will fit in the tight space between my walls? If anyone knows the answer to this question, I would GREATLY appreciate it!! Thank you!!!!

I would guess that you can use a ceiling fan brace kit in a wall too. Should work. Less than $20 at home centers and stores that sell ceiling fans.