I have no experience other than using naked lipos and separate (from the project circuitry) charging solutions.
But I have seen, and a bit of casual googling finds
I just searched for
protection charge discharge lipo module
and there are others. I don't see a way to adjust the charging current with the one I linked above - part of making lipos last is two charge them gently if you have the patience...
As for the batteries themselves, again all my experience is with flight batteries which typically have nothing but the lipo cells. These are available at all sizes and voltages and power capacities.
Go to Amazon and search there, for example
2s 2000mah
turns up a nice variety.
HTH and be careful.
a7