OK. Rebuilding the board is probably past my expertise. Well, it does only have 7 components. So I guess not impossible, but as you said, safety is important.
Most likely unfeasible as you would not have the right materials/tools.
I know I might sound stupid asking questions when I really don't have the knowledge to know where to begin, but... what prevents me just bridging the surface traces? There are 3 feeds across the rear of the board for example. Top, middle and bottom. Top appears to be connected to nothing. Middle and bottom seems to be the positive and negative lines?
Only way to learn is to ask questions and study.
There is nothing preventing you to repair the traces from one board to another.
This must be done in correct way for safety and functionality reasons.
As an aside question. Where do they keep temperature monitoring on these things? I only see the positive and negative terminals on the actual batteries...
There usually is thermistor in suitable location (if it has one)
The third option is finding an existing board that does two cell charging? But I'd not know which parts shop/hobby shop to search for that kind of thing. Something like: https://learn.adafruit.com/multi-cell-lipo-charging/simple-balance-charger (Though that won't fit in the tab without modification ).
You cannot just use random charger it must be suited for your application.
Now to repair.
You can repair the two traces under the board by bridging them with wire (use suitable gauge)
First look into structurally repairing the actual board.
Do a layer repair on the non trace side.
Use suitable non conductive material to support the pieces in this fashion _____________________ _____________________
______________
Use a suitable epoxy (not 5 minute ones as they are rather weak)
Also do not clue the broken board together as the traces between can then cause potential problems leave 1-2 millimeter cap filled with epoxy
Do not be shy with epoxy long as it still fits the case.
Now when board is structurally good you can move to next step.
1.Reveal the trace by scratching the solder mask make sure you get good clean trace of at least 3mm so you can solder the wire on it.
2.Clean the trace with IPA or other suitable chemical before tinning it
3.Twist the wire and pre tin it.
Now all you do need to do is tin the wires between the boards best use some hot clue just in case wires get loose from soldering points and cover with some heat shrink.
Test connectivity with multi-meter and run charging test monitor any problems or possible heating or other effects.
Remember repairs like this are potentially hazardious also problems with circuitry and batteries can lead to all sorts of problems from shortening lifespan of the battery to burning your house so monitor the behavior of repair etc.
Be careful be safe
Remember you do these repairs on your own risk !