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.