to retro fit a hand rail in that area, i think the best solution is to have a board cut, shaped and nicely finished that will fit where you want the handrail, so that you can hit all the studs, and be long enough to get at least 3 studs (in our trailer, 3 strud is all there is on the wall where the hand rail is, the stud at the bathroom door framing, the stud at the corner of the entry/hallway walls, and a center stud...
once this board has been fitted firmly, it can be removed if required to install the hand rail firmly to it, and then reinstalled.... or with the decorative but structural reinforcement board in place, just screw the handrail to it as per the instructions...
in your trailer, the studs may be spaced differently, but another option is to find the studs, then measure the center to center distance between them, and lightly mark the location of the stud from from knee high to chest high, using pencil or tape.... and then, as the hand rail will be on a slant to match the angle of the stairs, it may be possible to find a handrail about the distance needed to be able to attach to the studs on an angle, of which an inch or so one way or the other can be adjusted to fit by changing the angle of the handrail slightly.... this may be a quicker install, but the handrail will only be as strong as the weakest wall stud....