I am already using SWFObject to load the flash piece on our homepage to users with a specified version. So I figured I could hack that code a bit in order to pull the version it was already determining for me, and plug it into the Omniture code.
After mulling through the SWFObject code, I finally was able to run this page successfully, which will alert the exact version of flash I am using:
Basically, you can use deconcept.SWFObjectUtil.getPlayerVersion().major to get the major version, etc.
Once I got that working, I was able to use that same concept to plug it into my tracking code.