I ended up opening a support case with MS. That did not yield a solution as they were not able to determine that the system database upgrade step never happened even though it was clearly missing from the logs while it existed in the logs for the successful instances.

They contend that the patch was successful regardless of the 20 - 30 procs that didn't get updated on the instance. The patch could not be successful if the instance failed over before those updates would have been run and that appears to be the case.

I'm still looking for any useful suggestions because at this point, I have just manually upgraded any objects the patch would have updated that I could confirm were different between a successfully upgraded instance and the failed one.