So, it have nothing to do with Your problem. ho ever, warnings about date and time *may* be interesting (most unlikely, but better safe than sorry), so of course post it here.
Still, I think You got charging chip damaged. Probably, most convenient way for You is to mod Your cover (or buy mugen cover and mod it) to contain charging port, + buy cheap small Li-Ion charger (most likely, for small AA-like cells) and mod it to have cable fitting into Your cover new plug...
By the way, writing this reminded me about some thing. IF Your bme isn't working correctly, how device know when to shut down? It shut down properly (bright white diode fading etc, @ little less than 3200 mV), or is it used to ~2800 mV and just die due to low power?