How do I get my 96 year old dad into our car by myself in order to get him out and about?

Some states have dial a ride for the elderly where the caregiver can go along with the clients maybe check with his insurance company sometimes they have a program or volunteers that will help with transportation. If your father is military Veteran they will pay for a lift for newer model vehicles you could check with VA or VFW.

I did get some PT/OT for someone with part of the reason being that they could work on getting elder in/out of a car. PT came out and did assessment. While they were there, I inquired about the car thing and how to get elder in/out. PT was flabbergasted that I'd asked something like that and her response was "You have NO idea how complex it is to actually teach someone getting in/out of a car safely." I guess she was right - I had no idea. PT did work with elder on basic tasks and strengthening, but never got anywhere near the complexity of getting in/out of a car because elder did not make progress on the simpler things, so she was unable to progress to the difficult things. Of course, part of the reason she did not make progress was that elder would not practice her exercises in between visits and then she cried a river about wanting more therapy so she could travel by car. Also believed she would have been fine in a car even without therapy if we would only let her "try" to get in the car. She also felt she could likely drive (haha), but we didn't let her attempt that either. Just know that therapy visits don't go on indefinitely, so unfortunately therapists do have to make rather quick decisions regarding what skills to work on and what things might not be possible anymore. Some people accomplish a lot with PT/OT, but others don't get as much out of it.

You have the spirit & confidence to try new ways - impressive. He is very lucky to have you as his carer!

Can you get a home PT or OT visit? This was arranged for one relative when I was having trouble & it really helped. Worked out what was still possible.

I must admit, since then, I have given up transporting two relatives in my car as they need more than I can do.

Now it's a 'walk' outside in a wheelchair or wheelchair taxi.

I do not know the answer to your question but I admire your willingness to tackle such a problem. I’ve never even heard of most of the things you’ve already tried.
If it were me I would contact his doctor and ask for an order for physical therapy. The therapist really helped me understand how to help my parents getting in and out of a ca. Neither had the challenges your dad deals with though my mom did have a fractured back once and we did manage.
You’ve inspired me to take my aunt out more.
I hope someone has some great tips for you.