OK. I tried my default backup program AppMonster Pro to back up the app (Mass Effect Infiltrator). Then I connected it to a Windows 7 netbook and it connected via MTP. However, I was unable to find the backup folder anywhere. Supposedly it's under /sdcard but the drive shows up as "Removable Storage" under "Nexus".
So then I tried File Expert (as suggested above) and backed up. Same problem. The default backup folder was no where to be found when I browsed the folders while connected to the PC.
So then I downloaed Astro File Manager and magically I was able to see all of he backup folders when I ran Astro on the phone. Astro has a feature where you can Gmail an APK. No Drive support. But I emailed the apk to myself and then opened Gmail on the Transformer Prime and it installed and downloaed the 450MB of needed files.
The game runs insanely well. The level of detail is simply jaw-dropping. This will be my Mass Effect fix until BioWare decides to release their infamous DLC to fix the ending of ME3.

Thanks for all the help!
But any idea why all folders are not visible when connecting to a PC?