Mayo spends on and looks better after cooking but tasted fattier than it should, I was one and done after that experiment.
I'll just leave the butter out for a few minutes for it to soften. Or I'll just use a low heat on the pan and use the butter as the fat to fry the sandwich in, toss in a nob of it before flipping to grill the other side.
For greens I prefer a heartier, easier to assemble leafy vege like baby spinach or rocket. Lettuce elevates the bread unevenly and usually spills out during the flip. Even if some of the rocket creeps out and gets lightly toasted it taste great.
Adding greens to a grilled cheese may be a sin, venturing into BLT territory, but I'm trying to get all my food groups in here.