The problem was that the acrylic on the side where the motherboard is mounted was several millimetres off the case due to the motherboard rear panel cutout. There were no supports on that edge of the case at all. My solution was to take some of the leftover parts I had from my Mac Pro Mode, namely the motherboard risers from its original case and use some Weld Pro to cold weld them to the SGI case.
This is what one of the supports ended up looking like:
I put a total of two supports.
Then it was a matter of drilling the right holes through the acrylic rear cover and screwing them in with the Allen head screws that fitted them.
The two supports hold the whole side of the rear cover flush with the case now. There is still some work to do on the rear cover to make it perfect, however now there is no gaping hole on the side any more which should help with dust buildup inside the case.