How to combine a charger for LIPO battery with a boost step-up voltage converter?
Yes the SparkFun board is designed for exactly that. The limiting factor is the JST connector, but as long as that 6A rate is fort short runtimes you should be OK.

Test the temperature rise in hot ambient conditions, inside your enclosure at the XH wires and plug, holding 6A for say 10x longer than you imagine.

The other issue is the very low charge rate, if the cycling usage minutes per hour gets too high for too long, might need a higher mAh capacity battery to carry through the high activity times until the quieter recharging times.