Sorry to hear that reseating/cleaning the EEPROM didn't help!
I really feel unqualified to continue to comment, and hope that someone will chime in who has some experience with an SDR... for what it's worth, here's more wild speculation from my vast imaginary world
I don't "really" know what that $SDR$$.$$$ file is all about, but if we're dealing with "Dynamic Firmware" that gets loaded into the chip on every boot, then that data would need to be stored somewhere, while the unit is off. So, it could be that.
So, my guess would be, that the SDR writes the content of the EEPROM to that $SDR$$.$$$ file (after it successfully booted once - USB mode wouldn't count, I'd think), so it knows that it successfully booted off of that before, and can load it back in, if it needs to. Either on every boot, or possibly it could be a firmware backup, that it looks for when the battery fails, and the firmware data was lost on power-down . (in case it loads the firmware into the EEPROM once and keeps it there after shutdown, via the battery).
So, if that was the case, then it's possible that the SDR.FW file and the $SDR$$.$$$ file have the same content. But possibly in a different format (or the SDR.FW file could have some sort of extraction app in it, that reads the data from a different section of the file, and writes it to the chip - so, maybe not a straight 1:1 copy process).
Do both files happen to have the same file-size? If not, then they're probably either for a different purpose, or in a different format (or one includes that extraction app or something).
What I'd try (but there's a bit of a risk, in case the EEPROM can't handle it and might need to be replaced or re-flashed afterwards), would be to rename the SDR.FW file to $SDR$$.$$$, then reboot, and see what happens (if you're willing to risk bricking the EEPROM).
If you have an Editor that is used for development (e.g. BBEdit on the Mac... that's free. Not sure if that, or something like that also exists on PC), then I'd try to open it with that, and compare what's going on inside the files. (Or just try Textedit first... sometimes these files are simple text files).
Stupid question, though (...my tech support past taught me to always ask this stuff, no matter how ridiculous, sorry...) the USB cable is DEtached from the SDR when booting, right? I could imagine that it boots straight to USB, whenever a cable is plugged into the USB port.
If the USB cable isn't connected, and renaming the file doesn't help, then I really don't know what it could be, and should refrain from responding again.
Maybe someone on here would at least be kind enough to send you their $SDR$$.$$$ file, so you can put it on your drive. Maybe it would actually boot from that, if that IS a firmware backup. Then it would still point to the battery not helping to hold the firmware in the chip, though.
Where are you measuring that the battery supplies power to the board? Or to ask the other way around... I'd think that if you just hold the pins of a multi-meter to the positive & negative posts from your battery holder, then you'd get a reading, no matter if power is actually supplied to where it's needed. So, I'd follow the traces on the board, to see to what component it goes, and then measure on the pins that connect to that component, to see if voltage is still present. Having said that, though - I'm terrible with electronics, and have heard before that you can fry something, just by measuring with a multimeter in the wrong spot on a board. So... please try at your own risk... or ask someone who has a clue what to do or not to do with electronics before trying that.
Sorry... I think that's all I can contribute to this, since I just don't know the SDR at all (I understand that it's non-PC hardware and that there's likely no PC-style BIOS, etc. - but that doesn't really help with figuring out what might be wrong, sorry!)
Again, best of luck! Hopefully you'll get it working!