Sounds like a simple repair, much easier than water damage.
With only a 2" hole in the outside, from your description, I would temporarily patch the outside with a piece of tape and cover it with dicor so you don't have contend with keeping the rain out.
Then, repair the inside first.
What is your ceiling material? You can cut out the splintered area(Oscillating saws are great for cutting a hole to be patched), glue in backer material around the edge, and then patch it with a piece of plywood of the same thickness. Is it 1/8 or 1/4?
The last interior ceiling I repaired, I found a piece of plastic coated wall paper with a similar texture to the ceiling and then painted only the patched area. It turned out to match very well.
Insulate from the roof with expanding polyurethane foam and place a 1/16" piece of aluminum sheet 4"x4" under the rubber. The foam insulation will fix the plates position when it cures so no need for screws, just some weight until it cures and some support in the ceiling area to prevent a bulge from the expanding foam.
Then repair the hole in the rubber with either Dicor or Eternabond.
I'm a Dicor fan. The cost would be the interior ceiling, insulating foam, a piece of aluminum sheet, and the caulk you were going to buy to do the regular maintenance.
Post some pictures, might give everyone more insight into the extent of damage.
Good luck with the project.
Joe and Evelyn