Before I got a chance to apply my 10.2.0.4 patch, my problem fixed itself! Well, maybe not fixed itself, but certainly went away.
As I was reviewing my backup log from last night, I noticed that my RMAN-08137 error was gone. I didn't do anything different nor did I change anything. In fact, yesterday's backup had the same ORA-08137 error just as I had on the last 12 days.
So what was different? My control_file_record_keep_time was set to 1, so any record of my archived log would surely be gone way before now, right?
No, the control_file_record_keep_time parameter specifies the minimum number of days the recyclable data will stay in the controlfile, not an absolute limit. Sure enough, although my control_file_record_keep_time was set to 1, the latest log from my v$archived_log was from 12 days ago. All the records from the day I had a problem were gone.
Theoretically, I guess I could have switch my logfile a couple thousand times after a day and that would have cleared the issue up.