If you want it done right, do it yourself.
If you send it to Gryphon it will get done right - but I'm not sure if they do anything but over-the-counter repairs any more.
If you get on the Les Paul forum - I'm sure there are some good suggestions over there.
There are two critical steps in a refret. First is fretboard level, the second is getting the frets seated fully, and seated well. If you have an unseated fret, it will sound like crap, no matter what - even if its level.
So, get the board level, match the frets to the slots, and get the frets down into the slots solid. Some will argue that hot hide glue in the slot as a "solidifier" is a good thing. On a LP, I tend to agree.
Also keep in mind that the fret ends will be stainless - not the "nubs" of binding, unless you want to rebind the neck - or go the stupidly expensive route of building up binding material for each fret end and trimming it down. So, just know, it wont look "exactly" the same, and it will obviously be a refret, unless you go for a rebind and partial refinish.