Not that it matters much, but your fiber-glass Starwood looks to have a fold-out handle as standard?
It could also be that you need to angle the screws a bit towards the door frame where there will be a stud. You might even be able to notch the plastic bracket (that screws to the wall) to be able to move it closer to the door frame. I had to do that on a previous TT when retrofitting a larger handle.
If relocating the screw locations, use a pilot drill bit, esp. if the framing is aluminum. Wrap a piece of tape on the drill bit so you don't go too far.