Powering module directly from LiPo Charger with LiPo connected
I don't think this is good idea. The charger measures voltage and current to determine proper charging profile for LiPo. It cannot distinguish between too high battery current and normal current + additional load. Plus, in your case it is simply too weak to support combined current (max battery output is 1A).

That is the reason for modern BMS having "power path" technology, which does tricks like charging battery while feeding the system with extra power if it is available, or supplementing weak power source with some battery discharge. Your MCP73871 is exactly that kind of BMS! I'd recommend using it as intended.

Keep in mind, that


the GSM modules themselves are either 3.3V or 1.8V devices. No matter how you supply power to it, it will go through regulator.
I don't know what solar panel you use but I doubt it can supply 2.8A, which is combined max charging current and max GSM requirement.


One more thing. You said your charger provides "regulated 5V output". The MCP73871 does not do this. If external power provided it goes directly to OUT pin. It will be 5V from USB and 6V from solar panel. So basically, there are no "losses of the regulator" you are trying to avoid, except those in the GSM module that you have no control over anyway.

UPDATE:

If you draw enough current to drive supply below battery voltage (which is not hard to do with solar panel) then (and only then, per Kirchhoff) the battery will start discharging. At this point MCP will go into shutdown mode (see UVLO in datasheet) which will cut the input off and (hopefully) protect the charger. So, yes, you will get your surge current.

However in all other conditions you will have charge curves off and nice precise chip doing fools errand. All that to solve a problem created artificially by bad choice of LDO. I'd recommend using ultra-low LDO (ISL80103, LD39200, NCP59300) for as little as 0.12V drop and let the chip do its job.

UPDATE 2

Come to think of it, why do you need LDO at all?

In most cases the output voltage is at or below 5V. Only the combination of fully charged battery, bright sun and GSM in some low power mode will result in higher voltage, and even then only 1V over!

What you need is a voltage clamp. Something as simple as zener circuit below or a bit more complex with FET like this. Note that both have exactly zero loss under normal conditions, unlike LDO which will always drop some voltage, even if input is below regulation level. I even think you can find high power 5.05~5.1V zener with enough internal resistance to get by with just diode alone, assuming GSM module has permanent connection to output.