Your grill may have its own regulator. This is the problem from what I'm reading from your description....
What happens is that the Airstream regulator is reducing tank pressure from 150 pounds per square inch (PSI) to something like 9-11 inches Water Column (WC) for the Airstream appliances. This is what the Airstream regulator and propane system is designed to do. The quick connect port is on the low pressure side of the Airstream regulator...so you get only that very low pressure to the grill.
Your grill has a regulator designed to input 150 PSI or so, and regulate down to whatever the pressure the grill needs. That's why its not working properly off the propane port--not enough pressure to get your grill regulator flowing enough gas.
Your best bet is continuing to hook up to the tank directly. Otherwise, you need to know the output pressure of the regulator on the grill, and bypass it if it is in the right range to use the Airstream regulator directly. See if your grill offers a "low pressure port" accessory. If it does, get one...
If not, if the current grill regulator delivers 9-12 inches WC to the grill burners, you could take the grill to a propane supplier with a full-service shop, and get the correct parts to bypass the grill regulator and hook direct to the Airstream regulator.
You could do it yourself, but you best know exactly what you are doing--being in the center of a rapidly expanding propane fireball can ruin your whole day if you get it wrong...
__________________
Rich, KE4GNK/AE, Overkill Engineering Dept.
'The Silver HamShack' ('07 International 22FB CCD 75th Anniversary)
Multiple Yaesu Ham Radios inside and many antennae sprouting from roof, ProPride hitch, Prodigy P2 controller.
2012 shortbed CrewMax 4x4 Toyota Tacoma TV with more antennae on it.