I think you are making it overly complicated.
This is the hose you need. The female end attached to the output you indicated in your photo. The male end attached to the propane regulator on your grill.
Coleman 2000005042 8-Ft. High-Pressure Extension Hose 
Or you can get what I'm using for about $80 from Amazon that includes the tee in your photo, 12ft grill hose and the hose needed to use an external propane tank. This have everything you need to do what you want. If you tap into the regulated side and run a hose to your grill that has a non-removable regulator (like the Weber Q series BBQs), it won't work. You can not double regulate the propane going to your grill.
Sturgi-stay Deluxe Propane Adapter Kit for RV's, Campers, Motorhomes
