Ok, I just got back from a very brief vacation, so i decided to take a look at this thread again. Thoughts:
The top left picture of original post - the negative contact in the battery well on right side is obviously lose or been pushed around. The Litronix module is really particular about battery contact all the time, to both wells, as the thrid leg of that crystal has to be front loaded with some voltage. If you take the front off the module(stay away from the wirebonds, leave plastic magnifier over display if it stays attached), you will see the side contacts are all just pressed onto the edge of the board. You can scoot them over and clean the area with a small wire brush, but then the choices are(a)
solder which doesn't stick real well if the contact pads look like aluminum; or(b)
silver epoxy, which sticks decently but has very poor physical strength.
To test it out of the case. The contact that touches the case is either up at the 12:30 position(right above time command button) or at about the 6:30 postion. you can put a small clip on it and while holding two batteries in the well, touch the other end of the clip to the time command contact to test function. Unfortunately, the setting procedure for some of those modules requires two buttons held down simultaneously, so it become challenging to test those functions out of the case. Another really important consideration - many(not all) Litronix modules, like virtually all TI(Texas Instruments) modules, don't actually run until you have set them. This was so they could be shipped with batteries in.

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