Between those two places I don't think you'll be finding campgrounds, though there is a RV park at Expo Square and another at Mingo Road just north of I-244/US-412 that would not be too much out of the way.
Since I prefer campgrounds to RV parks, I think I would detour over to Fort Gibson Lake where the CoE has six Class A campgrounds and a few more primitive areas.
From Fort Gibson, you could approach Bixby from the southeast and avoid going through the middle of Tulsa.
Tom Test
Itasca Spirit 29B