T5?
You obviously have to go to TBIT to check in the bags. I do not know when (it was about T-3Hrs last I did this) Until that, you are clearly stuck groundside.
But you do not want to go to T5. DL has moved over to T2/T3. There are airside buses connecting these with TBIT.
You did not state how you have DL SC access. You might have access to the KE (ST) lounge in TBIT which would be a lot easier.