I've always felt LP takes some effort to make it go, mainly for soloing.
Try
a Hagstrom Swede (or most any Hag) if you ever get a chance.
I never could get into the Strat/Tele thing too much. I have a sweet '52 RI Tele, and I use it from time to time...but I never liked the neck, especially the smaller radius fretboard that many Strat/Teles seem to have. I'm looking to get another Tele, I want a Thinline, but it will probably be with a custom, flatter neck.
LPs often have what I perceive as fat/chunky necks...some guys love them, buy they feel a bit 2x4-ish too me, though overall, I can play on them easy enough.
Love my Hagstrom necks though...but then, I've been playing Hagstrom Swedes forever...

...and their RI models feel almost the same as their vintage stuff. Very easy on the hand, and super-fast/smooth.
It's a narrower and flatter-back neck, and the RI models have a 15" radius (pretty sure the vintage are the same). I find it just fits my hand perfectly, and every time I play my Tele, which has a C-neck and 7.25" radius...my hand hurts after 10 minutes when playing rhythm. For leads, I can move on the Tele almost as easy, though with the smaller radius, the strings end up being a bit higher so they don't fret-out...so it's not as buttery smooth as my Hags.
My hand obviously prefers the larger radius fretboards...but I can deal with different neck shapes/sizes easy enough.