After thinking about the problem for some time I figured I needed to look at the center of gravity in relation to the two mounting points for the strap. Looking at it now, the CG is obviously going to pull harder on one end of the strap than the other end when hanging in a comfortable playing position. I concluded I needed to make the CG as close to centered between the two strap mounts as possible. I figured friction between the strap and the players neck could tolerate some unbalanced CG.
I designed a new set of side plates for the guitar. These plates differ from the originals in a few ways. First, they're wider than the original plates. This reduces the contact pressure on your leg when playing. Second, they have a concave shape. This allows me to stably rest the guitar on when leg. The concave shape makes it significantly more comfortable to play for extended periods of time. Third, the new plates have more mass in the back. This shifts the CG further back and helps to mitigate the tipping seen in the old design. Finally, The strap mounting points were shifted to be more centered on the CG. I didn't make it perfectly centered, but the strap should hold some imbalance in the weight by friction on my neck.
I think the new plate design just looks a lot better than the old ones. I had more input from friends on what looks good / bad as well as ways to fix what doesn't look good.
The upgrade parts really did the trick - I can comfortably play the guitar while sitting or standing. There is still a small amount of weight imbalance when playing while standing, but it doesn't prevent me from using the guitar. The old design was mostly impossible to play when standing up. The bigger side plates did add some more mass to the guitar. The old design was about 10.1 pounds, while the new design is around 12.6 pounds. I think I need to make a new guitar from scratch if I want to drop the weight and get the right weight balance. I think the neck and especially the head really mess up the center of gravity because they are the furthest portions from the body. An easy improvement would be changing the fretboard material to something less dense than stainless steel (titanium or aluminum). Overall I'm pretty happy with the guitar in its current state, and I probably won't get or build a new one for quite a while.