Removing fake self stick backsplash
Just like you remove tough wallpaper really.

Get a spray bottle full of very very hot water/vinegar - I use a 1:1 ratio.   The heat and the vinegar will help break down glue.   

Start at the top of the backsplash.   You say there is no room to spray the solution - I understand.   If there is zero chance of the solution penetrating skip this step for the top.

Get out most powerful hair dryer you have and heat up the tiles.   Do the very top in small sections, pull slowly from corners as you try to keep heat on it.   

If you get a little bit of the top done, spray in vinegar solution very heavily so it runs down the wall behind the tiles leftover underneath.   Spray a lot and spray over and over.   If you think the spray is penetrating you can stop the hair dryer, and spray the vinegar and hot water every 5-10 mins drenching tile area.   I find that going three times through - after about 20 mins - the glue gets super weak and the tiles start peeling off rather easy.

Use hair dryer to help if they don't.

Note:  The other option is just to rip them down. Skim coat of drywall mud which takes almost no time and then mortar plus tile for new tile.   This is if you are retiling the same area.  I don't see the need for being cautious or keeping the drywall in perfect shape if you are redoing it.   If you aren't follow the above advice.