I suspect you're
GREATLY over-thinking this, and making it far more complicated than it needs to be.
You don't 'draw current at the same time as charging', just as with a car alternator/battery as you draw current it takes it directly from the charging source, and only takes from the battery if there's not enough from the charging source.
However, as LI-ion is more complicated, and easily prone to damage (or bursting into flames
) - then it makes sense to use NiMh, or even NiCd.