For awhile, HF had a Red and a Blue line of "Gasoline Generators." I looked at those, and I think one of the blue ones had a 30A "RV" receptacle. Back in that "Day" some of the blue were called Chicago Electric, then HF adopted "Predator", made them all Red, and the RV capability went away.
So, if what you have is Red, listed on "Gasoline Generators" at HF.com, has the Roll Cage, usually shown with a Wheel Kit... Jobsite/Contractor/Open Frame 120/240 generator. Can provide only HALF of its rating on any ONE circuit.
Notes: The biggest one, 8750 Watt, includes an "RV Adaptor" like you mentioned having. It's still using only ONE leg, but since the genny is so big, half will support a 30A load. Your 4000 won't.
Next Note: Under the "Invertor" category, the bigger one also comes with an RV Adaptor BUT BUT BUT a huge difference: The receptacle it plugs into, is THREE PRONG 120 VOLT already. All it's doing is adapting from Twist Lock 120 to RV 120 using all three terminals. That's Line, Neutral and Ground.