Regarding flying DL and TIAT lounge - I believe you have to be invited... i.e. you need to get a invitation card from check-in counter. If you need to argue your status and access, you need to do so at the DELTA counter. A few years ago, I was denied with PLAT status (flying award ticket in coach)
Also unfortunately, the DELTA counter opens within ~4 hours of a flight, so don't show up too early hopping to check-in and get in the lounge.