A lot less solder should be enough on those pads, as long as there is no cold junction under

Both LEDs glowing normally means that the charging circuit does not detect the battery. I haven't observed exactly the same behaviour, as there was either no charging at all, or normal charging all the way. The combination of red LED for a while, then going to both, is strange. My unit did not charge from certain adapter + cable combination, but for example the supplied cable + laptop is working well. It will not charge when it does not have enough supply current, however it only needs slightly above 500mA and I believe most adapters / phone chargers can supply at least that much, if not couple times more. Oxidized contacts or not entirely good connection will have the same effect, as it will limit the flow of current to some degree. The cable that we include has gold plated contacts so it should last long, but it also depends on what is on the other side.
Is it the same regardless whether Gecho is running (not necessarily playing, just in idle mode), or shut down?
It might be a defective charging circuit too but then it would probably not show red light in first place, I am not entirely sure though.
The connection between board and Li-Po should be OK as it does charge at least for a while. The built in protective circuits were never a problem either, at least with those two dozen batteries that I've tested.
I hope you can figure it out, if not, and it looks like a HW problem, it might be worth trying to replace the MCP chip, I can send you a spare one, or fix the board for you, if it's worth the hassle sending it back and forth.