Hot damn! I think I got it!
I reinstalled 3.1 from scratch, installed the Blizzard 060 libraries, slapped in my PII+, installed P96, went into P96 mode, attached the board, tested the resolution I wanted, rebooted, went into Screenmode, picked a high resolution, clicked use, the LCD monitor kicked in with the display, then I hit Save. Rebooted....and...and it didn't go into the low-res screen but kicked in on the LCD! I powered off and on just to be sure and it still worked.
So I don't know what the problem was. Maybe something in the order of how I installed things? Maybe I didn't utter the right incantations to the Amiga Gods...who knows.
I just know I'm happy right now.

Oh and as an added bonus to all of this, I put in a brand new PC power supply to run things off of.