Tune o Matic bridge
Ok.. I thought it might be a string snapping problem (which I had with my LP for YEARS) and I found a way better solution than the ball bearing bridges. If the saddles don't intonate properly no matter where to position them, it's possible the fretwork is off (not that likely) or the bridge isn't located quite right (also not that likely..) I don't know the solution to your intonation problem, but I can comment on the ball bearing bridges...
A lot of the LP sustain comes from the design of the bridge, and from the headstock angle. This is because when the string is pinched over a fulcrum, more vibrations are taken from the string and transmitted to the body at that point. This is just my theory, but with my experience it's all held pretty true. If for example you take a Les Paul and a Strat, and feel the strings above the nut when you strum the strings, you'll notice that the strings vibrate a lot more on the Strat than on the Les Paul because it has a way smaller string angle over the nut. The same thing happens at the bridge; I know a guy who works at a local guitar shop, and his guitar is basically all self built. It uses a tune-o-matic style bridge, and when he was installing it (and uninstalling.. and reinstalling, etc.) he played around with the position of the stop tailpiece quite a bit, and was surprised to see how much of a difference it made. When the tailpiece was closer to the bridge, there was much more sustain, and better tone. However, it snapped a lot more strings and was harder to keep in tune. Get the idea? Well when you use a ball bearing bridge, there's one more thing for the vibrations to pass through before they reach the body. You will probably have an easier time tuning the thing because there won't be any hangups between the bridge and the tailpiece, but because the string isn't held tightly in place, you'll lose a bit of tone and sustain. Another example: try playing something around the 12th fret or so of a JEM with tons of string bends with and without the nut locked.. You should notice a difference in tone, sustain, and feel.
LP's also always seem to be very easy to bend strings on compared to say, a strat.. I won't rant even much more than I already have, but the bridge/tailpiece combination and the headstock angle have something to do with this.