I believe the 'extend-a-stay' connects directly to the propane tank. That means it is before the regulator and carries high pressure. I have a valve like you've pictured. It is after the regulator and therefore carries the reduced pressure. The stoves and grills that are supposed to hook up to it do not have their own regulator. (at least mine does not).
Me~Jim, 44
DW~Patty, no comment
DS~Dave,19
DD~Megan,17
DD~Emily,11
200lb. English Mastiff~Gulliver,8
2005 Expedition Eddie Bauer
2002 Palomino Stampede 235 SL
Jim's correct, and I was very unsure after reading other threads on here that said the Quick Connect Line was already low pressure, and the Coleman's RoadTrip's regulator would mean double the low pressure and therefore not enough pressure to run the grill. But the helpful man at the RV dealer said no, the pressure is lower than at the tanks but not so low and that you needed the second regulator... Anyway, he said try it, what's the harm, if it doesn't work, bring everything back. But it worked great! Pressure was fine, I cooked pasta twice, bacon, pancakes, sausage...I haven't tried grilling anything yet. But used it all weekend and no problems at all. There are so many different fittings, etc. I say go to the dealer, tell him the trailer you have/connection, try some connections and hoses see what fits and put an adapter on a hose and you should be good. I'm glad I took a chance, now I don't have to buy the little tanks, and I didn't want to mess with the big tanks up front and have this long hose coming off the extenda a stay connection - I've got kids and a dog and that's just an accident waiting to happen...
I too, am a charcoal fan, but I used the same hook-up for a gas stove. When we purchased our unit a few ago, the Dealer switched the RVQ for a 3-burner stove from a Starcraft pop-up. We used it for coffee & other cooking. I still have a Coleman stove out by the charcoal grill to use when I'm making meals. I use the 16 oz. or a 5 lb. bottle with a "tree." Things work great for us. Good luck!