I'm guessing that the fiberglass on your Itasca consists of sheets of thin fiberglass used as an outer skin over a frame of some kind with seams along the edges and corners. Our trailers are solid molded fiberglass, frameless and seamless, so they don't develop the kinds of leaks
you are dealing with.
I also owned a Toyota Itasca mini-motorhome a number of years ago. My experience with leaks
is that once they get started, they keep coming back because the underlying frame is weakened. The problem with carpet is that if you ever do get another leak, it will soak up and hold water, and possibly pull down the wood paneling with it. I'm also concerned that self-stick adhesive is not strong enough for ceiling installation, nor designed for the temperature and humidity variations in an RV application. I don't think combining it with another adhesive will solve the problem; the weakest link will fail.
If you are just looking for something soft and attractive as a headliner, another option is marine hull liner. Scamp
has used a long-napped version over foil bubble insulation to line their trailers for many years. It's thin, lightweight, moisture and mold resistant, and quite durable. In a small, flat area as you describe, it would be an easy install. There are also short-napped versions available from marine suppliers in a variety of colors. Here is an example (just pulled it up for illustration; I am not recommending this company):
Marine Hull Liner¬*-¬*Marine Hull Liner - Rex Pegg Fabrics
You might check with a boat place over there in Lake Havasu
City. They'll know what you're talking about, and they might even have some samples or scraps you could inspect.
Scamp's version, which is a bit fuzzier, is here:
I'm not sure about the adhesive, but quite a few people here have done this project. If you decide to go this route, post a specific question and you should get some direction.