Thank you all for your replies. I dug out the old adhesive, lightly sanded the hull area under the rub rail, cleaned the area a bit with acetone, washed acetone off with water and then re-bedded with 3M 4200 and a series of c-clamps and a rubber mallet cushioned by cloth to ensure bumper was all the way on. Some of the adhesives that were suggested to me were not readily available in the U.S., but 3M 4200 seems to have done the trick and is not as permanent as the 3M 5200 I was considering early on. It seems to be a solid fix, but I will repost if I learn otherwise.
As an aside, I ended up cleaning the rub rail with acetone and then rinsing the acetone off with water. I probably removed some of the finish on the rub rail but it is back to brilliant white and I will probably hit it again with hull cleaner and wax.