No larger than 12x12? Yike. Others may have some thoughts, but aside from a DIY sub of some sort the only thing that occurs to me right off are the Sunfire subs ( Signature and Solitaire12 are 13x13x13, Solitaire10 is 11x11x12, and Junior is 9x9x9). I don't have any good feel for price on those, but it's not a number your wallet is likely to thank you for. There may also be some small Dayton subs at PartsExpress that could fit and would cost a lot less.

If we simply focus on fitting in the space available in the cabinet rather than sticking with the 12x12 limit, there might be a few other options. I guess you could even lay an LFM-2 over on its side and fire it through your cabinet's cloth, although you'd need to put a felt or foam pad (exercise mat material perhaps) under the sub to keep it in place and to protect the nice finish. It wouldn't work as well as the Sunfires would, but it's likely to do a better job than a little sub from PartsExpress even when placed in such a different orientation.