mounting a hand held shower bracket to drywall
Your idea is pretty good; however, I'd be concerned about the load on the fiberglass wall.

Try to get a sideways piece of 2x6 or 2x8 between the studs. If the pipes are in the way, you'll have to notch them a little. (Probably the easiest method is to set a partial depth on a circular saw, cut a couple of kerfs and lever the waste out with a chisel or even a sturdy flat blade screwdriver.)

If the pipes are too close to the plane of the wall and you can't connect one side of the 2x8 to the stud, then you could run an upright partial stud (also on the flat) from the plate to the 2x8s.