I've never even heard of anyone having so much trouble with a KLR. Are you 100% confident you're not paying for some sort of dastardly actions undertaken during a past lifetime?
I have listened to quite a few people speculating about what might be happening to a KLR coil or CDI when it gets hot. These discussions take place in response to random, frustrating issues like yours. In all cases (including my own), they were resolved, then forgotten, without changing out CDI or coil. Make of that what you will....but no matter how you look at it, your issues sound like fuel. Look for the simple solutions before getting into anything complicated. It's almost always simple.
Maybe you've already tried this, but I'd rig up a gravity-feed bag or bottle (it can be hung from handlebars if you're not prone to thinking about what would happen in a crash) full of clean fuel and try it out. That way you eliminate the vacuum petcock--a high probability cause for what ails you--and dirty fuel in one fell swoop.
Petcock bits and pieces go bad from time to time, and sometimes symptoms are intermittent. Fuel, vent and vacuum hoses get kinked or swell invisibly internally (<---take note). My air filter takes on water at irregular intervals when parked--not run--in the rain. You might also consider the way that engine heat is transmitted to carb and hoses. Some people have sworn by self-adhesive reflective foil tape to keep those areas cool during hot weather. Etc.
If all else fails, bring it to Javier. Or bring Javier to it.