I’m sorry. No, I don’t think it’s an issue with the battery or the SIM.
When you were running in Transparent, the XBee3 just didn’t register an internet connection. They just do that sometimes – refuse to acknowledge they are registered on the network. Sometimes switching from transparent to bypass and back to transparent “magically” fixes the connection issue on the transparent side. Do you know what version of the Digi firmware you have? The newest is a little better than older, but it’s still not perfect.
When you were running in Bypass, you were hitting the same issues that you hit with the UBee, except this time it connected, opened the channel, got data, and then somehow didn’t realize there was a connection and re-opened it several times.