Hopefully there will be test pads on the original unit for the SPI connections.
The lack of EEPROM on there leads me to think its a custom ROM for that particular headset/product. This may or may not be a deal breaker because it means the 8670's available on ebay and the like will have a different ROM mask/version to the one you are replacing. I don't think you have much hope of recovering the original 'settings', as they would be permanently in the ROM mask.
However. As there is a footprint for an EEPROM, there is hope. Any new device (8670) you'll solder on there will most likely expect an EEPROM - as all the 'modules' on ebay have them on board. Now, if you can find any test pads for config (might not be any as it seems the device has no EEPROM, and so, won't require configuration) you could use your programmer... but as you have to solder on an EEPROM anyway, you might be able to program that first, them solder it on.
I can provide a hex file for the EEPROMs I have on my modules. (I have an Arduino sketch for reading a dump somewhere..).
But again with the bad news... whilst there are 'standard' profile setups for these things (like 3 button headphones, 5 button headphones, single button hands free etc..) your headset could have a completely custom setup, with specific PIO's used... if you have traced out the PCB to those PIO's then you can set them in the 'headset configuration utility' that is kicking about the web.
So I suspect the reason it isn't up and working is one/some/all of the following:
-Not all connections soldered, a real possibility with BGA rework. No insulting your skills here!
-Different ROM mask, one that expects an external EEPROM to have settings. If none is detected, these things remain in standby. Good call on checking the output from its internal 1.8V regs.
-Wrong pin mapping from buttons to IO.
Given you removed an 8670 from a module, if the EEPROM footprint is the same on your headset.. then you could swap over that too. Hopefully the headset will use the default pin mapping for control buttons and LED's. If its a different foot print, looks like you'll either have to bodge it, or.. buy a new EEPROM.. clone the old to new, then solder that.
Sounds like mission, but it could all work out!