this is probably stating the bleedin obvious (so, sorry in advance) but G strings always will sound a bit weedy compared to your E string because there's less metal wobbling around in front of your pickup... and since the notes are higher in pitch you'll feel them in a different way.. more in your head and less in your stomach
like Bruce said, you can compensate by raising/lowering the ends of your pickups
on a Fender P your E string can often sound disproportionately loud anyway because of the curvature of the fingerboard... the radius is a lot smaller compared to many modern flat fingerboards... so when your action is set equally across the strings, the E string is closer to the plane of the body than the A & D strings...
the only other thing I can think of it that maybe your g string is set up slightly different to the others and it's rattling on the frets and being slightly choked... but since you said it's only noticable when you plug in maybe this isn't it... although the high volume could just be magnifying the existing problem...