The coleman voltage generator for 250 kva gen set

yes
the adjustment is by the rpm setting on the governor spring
thee is a nut that you adjust the spring tension with and lower the rpms and watch the voltage gauge until you get the desired voltage
if the load is going to be constant but is well within the rated capacity( example 2kva of a 6.5 kva set) of the gen set then the variation when the load comes on will be less than 5 volts
however if the power required is close to the max capacity of the gen set ( example 4.5kva of a rated 5 kva set) , then turn the unit on and adjust the rpms up to the required voltage required
remember that an advertised kva is not the rated kva
rated kva is what the gen set will produce all day and is normally around 1kva less than advertised kva
so if you have a 6.5 kva gen set ,it will handle current peaks to 6.5 kva but will burn out if required to run long time at that load
if the gen set is 1000 watts then rated watts will be less ( 750-800) for long time use

try adjusting the gen voltage at no load to run at 240 volts
when the load comes on it will drop back to 220-235 volts
check the load output from the inverter as 2.5 kv is nearly 80% of the genset duty cycle
4.5 KV is start up KV and the duty KV is around 3..8 kv not the plated 4.5 kv
to get constant 4.5 kv you will need a 5.5-6 kv unit
if the current draw from the inverter is excessive , the engine may not have enough in reserve to over come the load and will not pick up the rpms
take the gen set and the inverter to a shop that has a calibrated load bank to test accurate power output to determine if the engine needs an overhaul

If your generator has brushes you may need to "excite" the generator with a 6 volt battery

According to the parts pdf, there is no voltage regulator. Voltage is adjusted by the rpm of the engine. If your have a meter that reads hertz (Hz), you can set the engine rpm by adjusting hertz to 60 cycles (Hz). At 60 cycles the engine is running at about 3600 rpm. Most engines develop their max horsepower at this rpm. If you are loosing power/voltage and the engine speed is not dropping below 3600, than most likely could be problem with capacitor, brushes or diode (bridge diode). The diode converts ac voltage from stator to direct current to power the magnet (rotor). Check for brush wear and test capacitor for value printed on part. It should test with 5% of stated value. Diode can also be easily tested with meter. Good holidays

soud like an engine management fault
or a more complex avr does it have an electronc govenor

It should read 115v-120v with full load. Yes, you can slow the engine down just a little, depending on the engine, there will be a high rpm adjustment. 130v with full speed, no load, is really not too bad.

I’m happy to assist further over the phone at https://www.6ya.com/expert/rolandw_c013176d62ee17ae

set at 3600 rpms at 60hz

Engine speed is low, does your meter read frequency? Set for 60 Hz @ speed. I doubt there is a voltage regulator adjustment. This may help

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

Hi,
The engine should run at 3750 rpm at no load. If that's OK then you may need to troubleshoot the capacitor inside the end of the generator opposite the rope. Since the thing has been stored for some time you may just need to flash the field of the generator to restore the residual magnetism in the rotor. The easiest way to do this is thru the 110 outlet but it's a little dangerous. Plug an old lamp cord into the outlet, start the generator, and momentarily connect a 9 volt dry cell to the wires, then shut the generator down and see if there is 110 when you restart it.
Regards,
Carl

Both the Hz and the voltage will be wrong if the engine speed is low. You need to use either a DMM with a HZ function to set the engine speed so that you get 61.5 Hz at no load, or use a tachometer to set the no load speed to 3750 rpm.

Carl