Build yourself a testing rig for instantaneous switching between amps and guitars, or simultaneous operation of two amps. Something like an LS-2 or similar switcher and/or a pedal with stereo output capability should do the trick.
Personally speaking, I never feel comfortable auditioning new presets unless I've done comparison testing through my two main amps (a Marshall solid state, and Laney tube combo, both open back 1x12 with dissimilar speakers). I feel that doing the comparison through these two gives me a better idea of how it may translate through other amps, as they sound fairly different from one another, i.e. the solid state sounds more scooped and treble-heavy, while the tube sounds more mids-focused and "forgiving" in the treble/presence range.
As for the output levels of the guitars, I see it as a necessary side-effect of two guitars simply not sounding and feeling the same way. If it were simply clean sounds, you can try using some sort of mild compressor/limiter effect to make sure that both guitars would sound level and consistent every time you switch.
If we're talking distortion, it's a bit trickier, as the pickup output directly influences the saturation and feel of the final sound. You may either want to add some sort of booster pedal so that the strat can "catch up" with the louder humbuckers on your LP, or set either guitar up such that the output levels of the pickups are somewhat similar (strat pickups raised as close as they can get to the strings, LP humbuckers get lowered to match).