Hi,
No the GFI will not protect against a surge or over amp condition.
Whole house surge protectors should not be that expensive, and I do not understand them saying you need a 'larger' system when there are more circuit breakers to protect. The surge protector is sized based on your main panel circuit breaker size, not how many circuit breakers are attached to it.
As for the church, there is a problem that keeps burning up the surge protectors. Have an electrician check the ground bond, tightness of all the lugs on the major circuit breakers. At my work, there had been a 480 volt 225 amp electric panel in use for 15 years, and we discovered one of the main lugs going into that panel was just 'snug' and not actually tightened. My guess is the electrician put on this wire to this lug, and tightened it just enough to keep the huge wire from slipping out, while struggling to get the other wires on the lugs and tightened all of those wires, just not this one connection. Anyway it went on for 15 years! And the panel has a load of between 50 and 100 amps on it at 480 volts!
So yes it can happen.
When I worked for City of Long Beach CA, they checked all of the electrical panels with a temp sensing camera and found that many lugs needed to be tightened. Another review the following year found that everything was still tight. So I wondered if the original contractor employee never did tighten the lugs, or if they deformed over time? and heating and cooling as amperage goes through the wires.
Have fun camping!
Fred.