Update: I tried this experiment with the same V3 (however, this time with a regular CR2032 battery) and some interesting stuff happened.
But first, I have to get out of the way that "tilting" the battery while it was on the "CONNECT" screen did absolutely nothing. If you did it too fast, nothing would happen because the screen had a low refresh rate. Even if you did it at just the right time, it would say "FAIL" on the other one because AFAIK, IR data is transferred at an insanely high speed, so even tilting the battery within 0.5 seconds would cause a lot of data to be missing, which would make the other device display the "FAIL" screen.
Now, the interesting stuff:
One time, the screen turned off and it played the happy sound (the sound your Tamagotchi makes after winning a game or gaining points.)
Another time, it displayed some pixel squares on the screen in the usual glitch pattern but also displayed the "HOW MUCH?" screen you get shown when you want to donate to the Gotchi King and below it, the screen showed 1000 points.
That's basically all the interesting stuff that happened. I doubt I'll be doing this experiment again, because I don't want to brick one of my Tamagotchis.
(I also took this opportunity to ROM test it, and its ROM version is A3 3. Nothing special or new, but I thought it would be interesting to include.)