I have two Honda 2000i generators I usually use with a parallel device. When connecting the second one, its overload led comes on and it does not generate power. It starts and runs fine indepentently but does not generate power when tested with a small appliance.

pages 17 --19 show reset and overload lights indicator lights as well as trouble shooting guide https://cdn.powerequipment.honda.com/pe/pdf/manuals/00X31Z396030.pdf

There are several types of overload to consider:
A. Catastrophic: something has gone very wrong. There may be a phase to phase short or two shorts to ground. This assumes you have an ungrounded system. With luck the main circuit breaker will open without destroying itself;
B. Steady state over current: The load on the generator is greater than the rating on the circuit breaker. Time to parallel in another generatoror shed load;
C. Transient over current: Suppose you were running two generators in parallel and one failed. The remaining machine may experience an over current condition that causes the breaker to trip. You can also have transient breaker trips upon activation of a large motor e.g., fire pump or bow thruster;
D.Thermal: There is a big difference in sea water temperature as you travel the world. It is possible to have an overload caused by the diesel engine's inability to shed heat in the tropics. Continued operation could damage the diesel engine. Again, parallel in another generator or shed load.

No idea but somebody else will probably come along with a better idea then mine, seems a lot of these generators have this thing installed on them (Has $25 hard start kit). Not sure what it does or how it works. Strangely enough you said you use it for 240 volt. Looked at the picture and didn't see a 240 volt connection.
If you are using the what look like the 120 volt circuits you can't over load the unit. Maximum current draw will cause it to go into protection mode and shut down the power load. The following is an exert from the owners manual which I got from Yamaha's site. Link at the end of this post.

Hope this helps


-
16
-
q
760-029
AE00788
Overload indicator light
The overload indicator light
1
comes on when an over-
load of a connected electrical device is detected, the
inverter control unit overheats, or the AC output voltage
rises. The electronic breaker will then activate, stopping
power generation in order to protect the generator and
any connected electric devices. The AC pilot light
(green) will go off and the overload indicator light (red)
will stay on, but the engine will not stop running.
When the overload indicator light comes on and power
generation stops, proceed as follows:
1. Turn off any connected electric devices and stop
the engine.
2. Reduce the total wattage of connected electric
devices within the application range.
3. Check for blockages in the cooling air inlet and
around the control unit. If any blockages are found,
remove.
4. After checking, restart the engine.
http://www.yamahamotorsports.com/assets/service/manuals/0/LIT-19626-01-23_1423.pdf

Check to see if there is a breaker that has popped as a first diagnostic step

Make sure it isn't trying to start the HVAC with a locked compressor (IE it just turned off and is now trying to cycle on again) If it is shutting down due to low freq there is too much load for the HP the engine can deliver. That will cause too much current to flow and trip the breaker. Also be aware it takes 6 times current to start HVAC than to run it. How old is the generator? it may have normal engine wear and can no longer generate HP, Also when was it tunes up

It is a inverter protection activation (e-22). Usually followed by e-20,e-10, e-04. Could be a short the inverter or the gcu.

starving for fuel. the generator will always rev up when it is under a load or about to run out of gas. there will be a few adjustment screws on the carb... one for air and one for fuel. try increasing the fuel a quarter turn and see how that does

ARE SURE ITS THE OVERLOAD LIGHT AND NOT THE NOIL ALERT? IF ITS THE OIL ALERT ITS CAUSE YOUR LOW ON OIL OR THE SENSOR INSIDE THE GENERATER IS BAD?

Honda is pretty tight with the codes. You have to call a 'Authorized Dealer' to get it

If you need further help, I’m available over the phone at https://www.6ya.com/expert/craig_3fa289bf857b1a3c

Check the Ground Fault Indicator circuit breaker for triping, and look for the high limit on generator. Reset the main breaker by triping it full right and then back. Vitally