OK all I think I have a working prototype!
I say I "think" and call it a "prototype" since I have only done limited bench testing but it does work. Below are the details.
Mods required....
1. Must relocate the speaker. I placed mine on the top of the Micro SD slot. It's still plenty loud.
2. Must be willing to grind away the speaker mount in the case.
3. Must be willing to grind away some of the case itself to increase the width (top to bottom) just a hair. I did fine on the bottom but then rushed the top and now have a small 1x2mm slot on the top of the case where I went a little too far. I figure it's just some more cooling.
Unknowns....
1. Effects of GPS current draw.
2. Length of clock operation.
Configuration....
Rev1 - For revision one I used two 6F 2.7V Super Caps, Eaton Bussmann P/N: HV0830-2R7605-R, in series to get the voltage required, see picture below. This configuration was hit and miss. It seemed that sometimes the Mini 0803 would shut down OK and other times it lost power right before shutdown. The benefit of this configuration was that I could leave the speaker in place and only needed to notch the corners of the speaker mount to fit the caps. Although in the end it didn't work.
Rev2 - For revision two I bumped the total farad count to 9F
(<- Correction 4.5F not 9F) by adding two more 3F 2.7V Super Caps, Eaton Bussmann P/N: HV0820-2R7305-R. These caps were combined in series to get the voltage required and then placed in parallel to the caps above, see picture below.
The drive home today will be the real test since I'm not sure how the extra current draw of the GPS might affect the operation. This is due to the fact that the GPS mount is stuck to my windshield and I've only had time to bench test.
Although I can say in all confidence that if this doesn't work, the only solution would be an externally mounted capacitor since I have literally used all the room available and more.
I will keep you all updated.