Yes, measure voltage at the microwave with it off, and then with it turned on and see how much the voltage drops with a load on the circuit. Do the same at the breaker in the panel. Having a low voltage only at the MW recept. makes no sense and only explanation would be a bad connection. If it drops significantly and all other receptacles are okay, it's a problem in the MW circuit.
As dougranier mentioned, RV receptacles use a stab/push-on wire connection and it could be bad. I pulled a recept. out of the wall in a previous TT and found one of the wires barely hanging on. There's a special tool that should be used at the factories for pushing the wires down but it seems like the workers don't always know how to do it right. A loose connection could be a fire hazard too and should be fixed asap.
Go measure the voltage at the pedestal with some heavy draw loads running and see what happens. Use the 30 or 20 amp recepts. to measure. May not tell you what voltage on both legs is and would have to take panel cover off to check both legs. Low voltage in CGs/RV parks is very common and the staff are not licenced electricans. Testing a pedestal with no load is pointless. If your Surge Guard is dropping out, that would indicate the voltage is getting down to 102 volts which is very low. What is the plug on the shore power cord like? If it looks pitted and dirty, clean it until it is shiny metal. If you have a 50-30 amp adapter, try plugging into a 30 amp recept. Otherwise the only solution would be to move to another section of the CG that might have better voltage, move to another CG or get an autoformer for when you run into this in the future. In the meantime, do some load management and reduce loads where you can (turn off electric in HWT) and only run one appliance at a time.
Yes, it is possible to have two separate issues going on at the same time. If this is an older CG, they are more likely to have voltage issues. I'd recommend getting a plug-in or better, permanently mounted voltmeter so you can always see at a glace what voltage is doing.