The OP's built in quick disconnect outlet for grills, fire pits, griddles, etc is almost for certain going to be a low pressure outlet supplied off the regulator that is mounted between the two gas bottles. You do not want two regulators in line supplying the generator. If this is a factory installed system you can bet it is low pressure, but indeed, you need to verify that.
I have a built in Onan 2500LP generator that runs on the regulated low pressure gas of my trailer just fine.
All of the dual fuel generators or LP only generators come with a regulator that connects to the LP bottle, and a hose and connection to the generator.
The generator runs on low pressure gas. I have seen this question asked before and people who have dual fuel or LP only units say they run theirs off the trailer's low pressure gas system with no problem.
A 4000 watt generator will require less than 80k BTU at FULL load to operate, a standard RV auto changeover regulator such as the Marshall Excelsior MEGR-253 is capable of outputting 225K BTU on the primary cylinder and 150k on the secondary cylinder during auto change over, if you manually switch it, you regain the full 225k output. Even the low capacity model MEGR-253L is capable of 175K on primary and 100k on secondary. I installed (needlessly) a MEGR-253H which is the high capacity model, capable of 350K on primary and 200K btu on secondary. This is fairly representative of regulator outputs be they ME (USA made), Fairview (junk in my opinion) or Cavagna (Italian made with a very visible change over flag system).
Basically you can run everything in the RV and still have enough btu capability left to run the generator too.
Edit: found a discussion on a Forest River forum
Dual fuel Champion Generator - Forest River Forums and the OP said that he called Champion and they said .5 psi after the regulator which is basically the gas pressure in your camper's system.
One quirk is that the generator has a female quick disconnect and the hose needs a male qd plug on it, thus you would need to build up a hose with male qd plugs on BOTH ends and use the shutoff valve on the trailers disconnect, and be certain it is shut off before disconnecting any hoses. This is kinda like using an extension cord with male plugs on both ends, not real safe unless you are aware of what you are doing and carefully follow the process.
Charles