Here is what I know and/or think: Rematch trainers don't rely on the internal battery, but randomly choose to want a rematch when you enter the same route as the trainer(s). The TVs always plays something and resets every time you leave the building, which means they don't use the internal battery. The in-game clock only uses the internal battery when the game isn't being played.
Other that that, I have some ideas on what maybe the case for this question.
1. You may have bypassed the "internal battery has run dry" message before you start the game.
2. You some how changed the internal battery and now died or has very low juice and the message failed to appear.
3. The "internal battery has run dry" never appeared , therefor failing to notify you or anyone else that the battery is dead.
4. The battery is charged, but some glitch is messing with it.
Anyway, check if there is a message at the start of your game, if so, that's the case. If not, it could be one of the other mentioned problems. I hoped that I helped.