Most important imho, is how it feels, plays and maybe, looks
For instance:.
When i went for my 2nd guitar, i thought to myself, i'm going to buy a Gibson SG or a Telecaster. Done. Nothing else. Untill i got to play them. They didn't do it for me. Didn't feel good. It just felt wrong.
Untill the store assistent got me an Ibanez AS-153 and boom, instant chemistry. I knew that was a guitar for me.
What i want to say is, don't get too locked into a brand. Visit some stores, try several guitars, and see, hear and feel what you like. See which one connects, if that makes any sense.
If it's important, you might want to consider bridge type, if there's single coils or humbuckers present (maybe even which ones) or both, neckshape and width, radius etc... But that boils down, for me that is, again to how it feels and plays in your hands.
Making a sound like an LP is all down to pedals, amps, etc.... Clean, they all sound somewhat the same.
I hope i made any sense and it's helpfull to you. Good luck in your search! Let us know what you came up with.