More often than not, when the new connections show the "dead battery" screen, the battery isn't actually dead. Many people have reported that using a different brand of batteries usually works, but it also might be an issue with the battery contacts not making proper contact with the battery.
I had this issue on my rainbow sky shell earlier (only 2 days after changing the battery!), and after pressing the reset button and removing the battery and reinserting it didn't work, I tore up and folded a small piece of paper to put over the battery before screwing the hatch on, and that seemed to work. Maybe try doing that to see if it works?
Also, weight has no effect on your tama's health on the connections; in fact, having a high weight makes it much easier to win at Bump.