Charge lipo battery while using
You really shouldn't charge a lithium battery whilst discharging it. The charger may either never know that it has finished charging and overcharge it, or shut off early and never charge it up properly at all.

Instead you should arrange that your MCU is powered directly from the PSU so that it always has stable power, and feed that same power into a normal charger via some form of electronic switch so it can be turned on and off. The output of the battery can then be fed through another switch into your heavy load. 

When it comes time for your load to be activated you switch off the power from the charger and switch on the connection from the battery to the load.

What you use to do the switching is entirely up to you - whether you choose P-channel MOSFETs or relays entirely depends on your situation.

Here's a block diagram to better illustrate it to you:







simulate this circuit &ndash; Schematic created using CircuitLab