I have found a hard-sided cooler for the sub-trunk.
I went through
this table of coolers and found the largest hard-sided coolers that met the height limit of the sub-trunk, which seems to be approx. 11.75". There were 2 that came up:
Stanley Adventure 16 Quart and the
Mammoth Cruiser 15. The Mammoth is the more upscale brand, but the reviews on Amazon didn't give me confidence it was worth the price premium over the Stanley. So I ordered the Stanley, and it fits perfectly and allows the sub-trunk cover to seat fully.