Success! Nothing a pair of wirecutters, a couple pieces of paper, and a whole lot of electrical tape couldn't solve.
The drive cage has a couple brackets used to screw it to the case. I took them off of the cage and used them for the optical drive, which fits when the brackets are screwed in upside down. There's a small metal bar on the bottom of the bracket that prevents the tray from opening, so I snipped the bars off with a pair of wirecutters.
Next, I just had to attach the optical drive to the brackets. The holes to screw together didn't align at all, so I went with electrical tape instead. I used one piece that goes behind the drive from bracket to bracket to keep the drive from getting pushed back into the case, and then a couple winds around the brackets to essentially compress them into the drive. The optical drive was actually ever so slightly smaller than what was needed to align the brackets with the screw holes on the case, so I used a couple folded pieces of paper on either side as wedges.
It's not pretty or high-tech, but it works. I probably could've gotten a much less janky result if I'd drilled some holes in the brackets to attach them to the optical drive instead of relying on compression.