Years ago, at a sound check, I tried using a Marshall Silver Jubilee amp with the speaker cabinet aimed at me like a monitor cabinet. I left the amps knobs set just like I normally used them, so my sound in the PA would be consistent with the sound that was usually sent to the mains. All the warmth, creaminess, space was completely gone. All I heard was dry, harsh, and bright. And, that was WITHOUT adding a microphone, or profiling into the equation. Aim one of your amps at your head - you will experience the same thing.
You can use ST or LP rigs with either guitar, if you like the sound you get. Most of my guitars provide both single coil and humbucker sounds, and I don't necessarily switch Rigs when I switch pickups.