Some say even new LP's don capture that LP tone. Opinions vary. I have an 81 and 2012, they do sound different a bit, Motor City's in the old one. Not gonna get in to older = better BS. the new ones sound great and play sweet too.
I have never owned a super Strat that sounded like an LP, nor heard one on a disc, live, or at a jam session. If it existed there would be a long line to get them. I have a Suhr Custom Standard and Anderson Pro Am. The Anderson plays like a dream. Does not sound like a LP into a Marshall.
As mentioned earlier, I believe the body shape contributes to the tone. Check out a used G&L ASAT Deluxe. I found one obscenely cheap used. It has a mahogany body, maple cap, maple neck, rosewood plank, Duncan 59 & JB pups in it. Very LP sounding, not like a tele, which is the body shape. I'm thinking about putting some Motor City's in it for shits and grins. It comes with coil tapping switch too. It may work for you.