Makes no sense to me either that life clerics have heavy armor proficiency. "Battle Medic" is more accurate. Just give them any other skill or tool instead.
I think this is a fantastic idea. I currently use the gritty realism and lingering injuries variants, but I wish I had thought of this when I started my current campaign. I'm totally using this next time.
The problem is that the party fights the big bad rested instead of after a long series of battles with all his minions when they are nearly depleted of resources. Every edition of D&D (and other RPGs) will have this problem with poor encounter design.
If you're using milestones, they level up when you say so. If you're using XP, just give out less. In my game, I give 1/10th the normal amount of XP and require several weeks downtime training. So levelling in the middle of a dungeon is impossible.