These generators don't change speed when running. Or at least not more than a very slight amount. It's not enough to really notice compared to the increasing amount of noise from the extra work the gen is doing with a heavy load. The volume gets louder, the tone deepens a little, but the frequency of the noise doesn't do much.
The 3 minute wait at startup is probably a function built into your charge controller to allow the generator to warm up a little before being put under a load. There will also be a cool down period at the end where the controller lets the generator run for a few minutes to make sure it's cooled off enough to safely shut down.
If the generator is chugging along not dropping in power (have you looked at the HZ meter? multiple that x30 to get engine RPMs) then the generator is doing it's job properly.
Basic lesson in generators.
The generator head will put out the amount of power the load is trying to draw. If the load is trying to draw too much power, then one of two things will happen. Either the engine bogs down because it doesn't produce enough power to keep the electromagnetic fields in the generator spinning against the resisting field created by the load drawing power, or somewhere in the system a wire is going to get hot from too much amperage going through it and you'll melt that wire, creating an open circuit through which no power will flow, or two wires will melt together through the insulation around the wires resulting in a short circuit.
If your unit is producing the required voltage, and is running at the proper speed to produce 60hz power, then what you are talking about is not a problem with the generator. The generator is likely functioning how it's supposed to.
Reading through the manual for the Conext SW charger/inverter it looks like AC input is limited to a 30 amp breaker, or 7200 watts. You never specified what model of equipment you're running, so I'll go with this one for now.
(When you say 5500 watts input, does that mean 5500 watts total, or is that 5500 watts going to the batteries to charge them? If it's 5500 total and not just to the batteries, you can disregard the next part, as it's mostly speculation about what could be occurring)
So maximum your system (if it's this model) would be able to draw is 7000 watts or so before tripping the breaker. That 7kw is going to go both to charging the batteries and bypass the inverter to output to your house/load as well. If your house/loads are drawing 1500 watts of power, then the battery input at that point will be 5500 watts or less. It doesn't convert all the generator power into DC power to charge the batteries just to convert it back to AC again.