It's impossible to get that classic LP sound out of a guitar if you don't build the guitar exactly like a LP and with the same type of wood. I also like the sound but it's not my favourite. Great sounding guitars
As for the tuning stability, that has nothing to do with the construction it has to do with the craftmanship. Most of the time (almost always) when a guitar stays poorly in tune it's because of a poorly made nut. Making nuts for guitars is a precision job which can be very difficult. It has to be filed and polished exactly right to get the guitar the best it can be. Unfortunately not even really expensive guitars always have a good nuts, I have a lot of experience regarding this, I've tried many expensive guitars with poorly filed nuts. This is because it is difficult to do, if you quality control lacks, bad items will pass through. But is has nothing to do with the construction. If Gibson have problems with this, it has to do with their quality control.
The bridge is fine and the tuners is fine, string it up properly and be sure to leave it to a skilled luthier if the nut is bad and it will be a great guitar to play.
With all that said a guitar with a neck angle and the scale length of a LP will always be harder to intonate on the higher frets. The longer scale length you have the more accurate intonation tend to get. But no one is complaining, people have played LP's for decades, the sound is amazing, it's all in the fingers anyway, how you intonate.