I know that this will be considered some hardcore necroposting, however I thought it would be interesting for this never-to-be-answered question to actually find its solution.
A brief comparison
I happen to have both a 880GM-LE and a 880GM-LE FX. Interesting thing is, at appears to be the exact same board. Not just similar physically, but exact same. The 880GM-LE FX is an identical PCB to 880GM-LE, down to even being the same revision, and the "FX" model name is just a paper sticker over the original 880GM-LE name. The same goes for the "AM3+" text, it's just a paper sticker over "AM3" printed on the PCB. Of course "8 core CPU support" is also just a paper sticker.
The photos on AsRock website for the non-FX version are bad for comparison, since they're of an old revision. Just imagine the non-FX board being identical to FX version but without the paper stickers and a couple more cap spots populated on the board.
Yes, the only visible difference on the pcb itself is cost-cutting, as they cheaped out and didn't install some of the electrolytic caps in the VRM section on the FX board.
The actual answer
So, does the FX BIOS work on the non FX board? It absolutely (almost) does! However, I had to flash it using a flash chip programmer rather than software, for reasons mentioned by previous posters.
**BUT** Why "almost"? It boots and recognizes itself as the 880GM-LE FX board, but only if you have an AM3 CPU installed.
It works only until you put in an actual FX CPU. Then the fans just spin up, and nothing happens, no beeps, no video. Dead.
That was quite unexpected TBH! After all it's the exact same board, with the only visible differences being paper stickers and a different bios flashed into the chip. I really doubt more or less electrolytic caps would affect compatibility in any way, if anything I'd expect the FX board to have more caps, not the other way around.
My current hypothesis is that perhaps the non-FX board has some of the socket pins not soldered? Some small caps underneath the socket are missing maybe? It's the same PCB mask so all the traces themselves should be there. This could be an answer as to why AM3+ CPUs were not compatible with AM3 despite initial information than they would be: Motherboard manufacturers might've cheaped out on the boards, not making them to full AM3 specs and only connecting things that the currently existing CPUs needed.
Currently I'm unable to investigate this further, since the FX motherboard is still being used in a family member's multimedia PC, but who knows, after another several years I may come back with the definite answer to this question.
PS. despite the crap VRMs the board has been (and still is) running OC'd FX-6300 for several years, so perhaps we're misjudging a bit what kind of VRMs some CPUs actually need.