Using a lipo charger AND discharger at the same time
Generally this is a bad idea.

You can work it that way but it can cause problems for the charger. The charger will think that the current being used by your boost module is going into the battery. This may result in the charger changing the supplied voltage.
I can't think of any way you could get into a dangerous situation but depending upon the difference between the charge current and the system power draw it would be quite easy cause the whole system to brown out, prevent the battery from fully charging or create very long charge times.

Generally if you have external power connected then you want to run your system from that source directly rather than the huge efficiency hit of stepping it down to battery voltage and then back up to 5V. Exactly how is best to do this depends upon the input voltage you have and the 5V boost module you are using. Some have external power inputs, some have a shutdown input. If the one you have doesn't have either of those then what happens if it's output pin is at 5.1V? Does it effectively turn off or would it connect 5V to the battery and damage things?