From the picture, it looks like there is an O-Ring on just behind the face, which would make me tend to think that the clear face and black back unscrew.
While not the same, I have an old Sherwood bottom timer that I have had apart for battery replacements many times, and that continues to work.
If it won't work after a battery change, how did it work in the first place? That statement frankly makes no sense at all . . . Especially considering that this was back when software on stuff like this was mostly regarded as a bad idea - all implementations were in hardware, so what is to be lost on a power failure? Worst case, find the reset contacts to light it up again . . .
Myself, I'd give it a try as well. Absolutely nothing to lose . . .
- Tim