I haven't read through the thread beyond the first few posts, but here are my thoughts.
I would try to talk to your group about experience points and why they dislike the DM simply telling them when to level up, for example once every few game sessions. I think that's the best solution - according to my playstyle. And I'd try to convince my players accordingly if I were in your position. Here's why.
D&D is a RPG. Giving XP only for battles means the players will be looking for ways to always fight. I don't want that in my games, quite on the contrary. I want that one hour where the PCs spent time in the tavern of some back country village RP-ing to get some information out of reluctant towsfolk, to be as important as anything else.
Generally, I don't feel that a numeric reward in the form of XP for successfully battling opponents, gaining information or otherwise moving forward in the story that unfolds, is necessary for the game to be fun. Quite on the contrary, again. I think it is detrimental to the fun. I prefer a game session where the players will decide on actions for their PCs, simply based on fun around the table and some kind of logic with respect to the actions of their PCs and what they want them to achieve. I don't want their decisions to be warped by a meta-rule that awards XP for certain actions (e.g. for battles only); or worse, for certain decisions (e.g. parley instead of fight).
The purpose of XPs is to track the evolution of the PCs through levels. We all agree that PCs will level up because everyone likes PCs to gain new powers and battle new more powerful opponents. The question is only, do you track XPs with specific parameters? Do they gain XPs only when they defeat monsters, only when they accomplish X action, or only when they accomplish X action in a certain way? Indeed, some might say: you can award XPs for clever RP or for interesting in-game ideas, such as trying to pay a round of drinks to the tavern clients. I will reply: who am I to decide which action is more worthy than others? This is a slippery slope I'd rather keep out of. Ultimately, maybe the barbarian's decision to punch the arrogant bartender in the face, will mean more for the fun of the game and for the story, than paying a round of drinks.
In my game, the story isn't pre-set. No railroad. The players' decisions will dictate importantly where the story goes. What action is worth more than another, is simply not aligned with my way of playing. Any action, be that parleying, punching in the face, sneaking around, casting a divination spell, paying a round of drinks, is noteworthy, and who knows what that action will lead to. Maybe the bartender will become a long-winded vilain in the story arc. The story is simply the result of the decisions that are made. Why am I to award more XP to the group that decides to parley instead of bashing their way through a group of important diplomats, thereby causing war to erupt between two countries? For the sake of the story and the fun around the table, each avenue seems pretty interesting to me.
Consequently, I think that XPs are detrimental to the goal I wish to achieve according to my playstyle, because they predefine what actions are worthy of reward.