90% of general tuning issues people bring me and blame on nuts, bad trems, cheap tuners, etc, are due to loose parts. Mostly the nut around the peg and the screws that hold the pegs tight.
With a LP type, you have a huge amount of down pressure due to the angle headstock creating friction, and the string going over the nut then shooting off at an angle to the tuner.
You mentioned you have a bone nut. Bone is fine for acoustics where you aren't bending strings all the time, or (maybe) straighter string pull guitars like a Tele or a strat if you can tolerate either using lube (yuck) or some minor tuning issues. On electrics, I don't recommend, nor do I personally use, anything but graphtech, and only use bone after a discussion of the above with a customer and at their request even knowing the potential issues. A properly cut graphtech nut will never need lube, and will stay in tune as well as any other guitar when used on a LP.
Another issue you have with a tune-o and stoptail type guitar is sharp down angle after the bridge over a sharp metal saddle. Most people see the screws on the tailpiece and assume "hey it's a screw, lets screw it all the way in!" but that is not necessary and can lead to tuning issues, collapsing bridges, etc. It only needs to be down far enough to create enough down pressure over the saddle. Also, the string going over that sharp, pointed metal saddle and being bent so it slides a bit can be an issue. Again, graphtech saddles are the cure for that, and I find they sound very nice on a tune-o type bridge.
If your bridge is loose on the posts and can slide a bit, that can also create tuning issues (as well as intonation issues as it moves around over time). Tonepros bridges are the cure for that. I don't think they do anything for "sustain & tone" as they claim, but they DO keep parts from sliding around and from falling off during string changes, and from having to re-do your action height after string changes. A fairly cheap investment, or get a tap & drill and some set screws and make your own.
A final issue is string stretching. Few people seem to stretch their strings out enough. I grab them at the 12th, yank them up and down, back and forth, then re-tune, and when they no longer go out of tune, they are stretched.
Even my LPs and similar guitars with trems (bigsbys, deusenberg les trems) stay in tune great, but it takes some extra steps to get there due to the non-straight string pull past the nut and the down pressure from the angled stock.