That sounds about right. It wasn’t designed to operate without a battery, or to pass through power for a high current system. Only charge and protect the battery.
Quite possibly, the max power LilLiPo will take from the power input is what it is giving the battery (212ma for that model). Any load on the output takes away from what would charge the battery. If your total load on the system exceeds that (which is quite easy to do with LEDs), you will likely be discharging the battery, even with power applied, since the battery has to make up the difference.
Yes, with a low battery the charge current is reduced to bring the battery back up to safe charge levels, and if a load is still connected, it will likely not be enough to power anything.