Mounting a toilet paper holder in any bathroom is always a problem.
On our bathroom remodeling jobs, we like to use recessed toilet paper holders, they come in a kit with a bracket for mounting in a hollow stud space. If you cut the square hole in the wall and install it properly you cannot easily rip them out of the wall.
Becasue of the small toilet rooms, a hollow wall the recessed holder is the best, and therefore, I installed a recessed holder in our Bighorn.
The wall material are very thin, after marking the hole on the wall (24" up from floor and around 8" over from right corner), drill a hole in each corner, and then you can use a sharp utility knife to cut the square out. If the wall cavity is not deep enough (I don't remender), cut straight thru the other side of the wall (everything in that corner is hidden by the vanity cabinet) to get the holder installed. The recessed holders come in chrome or white at Home Depot.
If you don't want recessed, mount the toilet paper holder on a piece of deco 1/2 plywood painted white and then use mollies to mount (we always use construction adhesive also). I think that you can now buy the mounting boards for towel bars & toilet paper holders at Home Depot now.