First off, I'm sorry for your loss.

Hopefully, some others that have navigated the process will be able to jump on here and add some insight, as navigating the Veterans Affairs stuff is, complicated, and when my pop passed away he already had everything in place since he knew his time was limited (Agent Orange) and my mother and brother took the lead with that stuff.

Depending on his status after discharge and his record, he should be eligible for some funeral services through the V.A. https://www.benefits.va.gov/compensation/claims-special-burial.asp

You may also want to post this question in a few of the Veteran Affairs, US Air Force, or other US military based subreddits out there, as they may be better suited to help guide you.

Again, I'm sorry for your loss, and I wish you luck as you navigate the unnecessary bureaucracy that is the VA...