You can draw power while charging, it should just mean the charger takes longer. A full BMS is the best
approach if doing this, as it can detect both under-voltage and over-voltage on every cell and cutoff, and
arrange for balancing during end-of-charge period, and perhaps report the pack's state on a display (fancier
ones). The load connects to the battery. The charger connects to the battery, and the balancer also connects to the cell junctions. Most packs use a separate JST style connector for the charger/balancer,
you should adopt this standard for your home-build pack ideally.
You must have over-voltage, under-voltage and over-current cutoff with lithium packs (the former
is only needed for charging of course).
If you never let cells reach full charge the balancing won't really happen, which might be an issue if
charging while drawing power. A full balancing charge isn't needed every discharge cycle though.
Make sure the charger or BMS is a match for the chemistry of your cells. (most are 3.7V nominal 4.2V
full charge, but some vary significantly, like LiFePO4).