Adding base shoe in lieu of removing the baseboards is a viable option, but it wouldn't be my first choice. Don't nail it down into LVP (it should "float"), nail it into the baseboards.
As I said, I don't imagine you have elaborate baseboards (since you have mdf or particle board under the carpet I'm guessing you don't have an older home? Because usually the carpet is just installed right over the old wood floors without adding underlayment) but plaster walls and big old (dry) boards ups the damage potential.
I know painting sounds dumb, especially from a carpenter (me), but I've worked for a few very creative artists who opened my mind a bit (but just a little bit, I'm kind of a traditionalist!) about non standard construction methods. I've seen laminate kitchen countertops painted that looked good for years, one client in his restaurant made color copies on thick stock of a picture of a beautiful, colorful floor tile, and lightly glued them to the floor and put multiple layers of clear finish over them, and it looked great! And they were all 8.5x11 ;-).
So if someone wants to get creative, go for it.
Recap: your sticky tile idea is fine, but beware how imperfections in the underlayment can show through in the tile.
LVP is easier to install, as you can adjust as needed, and it bridges imperfections with ease. I install it directly over the subfloor. It does make a little annoying dust when you cut it (many are on an MDF core, which is essentially sawdust glued into a sheet). I cut it with power saws and replace the blades after, because it will dull them prematurely
Install base shoe or reinstall baseboards, whatever you're more comfortable with.