First of all, Disclaimer: try at your own risk!
I was looking at my "expendable" (and expended) FOB and it looks like there is no glue at the corner where the key ring would loop through:
So if I were to attempt to separate the FOB, I'd try to wedge something into the groove in that area. Like this:
Once wedged in, a small screw driver blade should be adequate to separate the two halves.
The battery is a Panasonic VL2020
, but I don't know about general availability. I doubt that the tabs would line-up correctly anyway. Your best bet is probably ebay. This gives you one to practice on.
With the board removed, unsolder the battery here:
Once the good battery is resoldered into the good board, glue the two halves of the FOB shell back together with a suitable glue. I recommend Gorilla Super Glue because it is a light gel and will fill voids better than a traditional cyanoacrylate (crazy-glue). You could probably use ABS or PVC cement too, but they could get messy.