You can do better on the amp, but I donīt think that is the main thing here. I am mainly playing on a Yamaha THR these days and those buggers have come a long way. Got a Tiny Terror too, but in my apartment situation, it is just too loud.
So, pickups. You will likely want something on the brighter side, if the LP does not happen to be very bright to begin with. I would say have a look at the Rebel Yell. It is basically made to work in LPs. Tight, upper mid spike, overtones, rocking roar,...carries itself from classic rock and blues to the extremes of modern metal. Since you have pedals that can go quite extreme you should be able to cover a lot of ground with it.
The neck imho is more metal than the bridge, so if you want solos with gain on it is spot on, but if you like the neck in bright and also be able to do more classic cleans full on other options might work better. Not that the RY neck does not do cleans very well, but it does not do the PAF type of cleans.