Can someone opine on our itinerary for Frankfurt to Budapest/Prague by car

This is a lot of driving. I never did that amount of driving in that time when I lived in Germany.
Do you know where the car can be parked when you are in those cities?
People can define seeing a place differently. There is much to experience at any one or two of those places to more than fill your vacation time.
Spend some time with the Rick Steves books to figure out what you want to experience on this short vacation.

With stops for gas and toilets you are looking at 30+ hours on the road.

Take about 6 hours off the trip and skip Budapest and Bratislava and use those 2 nights in Prague and Vienna.

Have you associated drive times with each leg of your trip? Have you already bought air tickets? Can you change them?

The brief stops in each location suggest you may be trying to cover too much ground to have time to experience much more than the interior of the car.

To me this looks like two trips - one to Germany and Austria, and a second to Prague and Budapest. If it were me, for the first trip I would fly into Munich and out of Vienna.

We already have the air tickets. Maybe we will cut out Budapest.

Prague, Budapest and Vienna are large, world-class cities. Two nights in places like that (really just 1-1/2 days at best) is just not enough time.

I don't know about the Frankfurt-Augsburg-Innsbruck trip segments, but otherwise it looks to me as if the car will just be an expensive burden for this itinerary. There are lots of interesting smaller cities that don't have super-fast train service in Germany, Austria, Hungary, Czechia and Slovakia--places you might travel to more efficiently with your own vehicle, but they are not part of your itinerary. For the trip you have planned, it looks to me as if the car will cost you money and time.

People rent and use cars in Europe to reach rural areas, not to drive between major cities. If you were stopping in between places, maybe a car makes sense but you are hauling butt to get from all of these places to the next. I have lived in Europe for 16 years and we’ve never owned a car. We take the train or public transport wherever we need to go. For this route you will save time, money and your sanity by taking trains. Also, you need to cut out some places.

Nothing wrong with the attempt to drive. Of the cities you want to visit, while Budapest is my favorite, it really stretches the other stops too thin. Don't worry, you won't be alone on the road as more than 75% of Austrian and German households own a car. I guess tge other probkem is that with your schedule you won't have the time to utilize the freedom a car will give you.

But for cost savings do look at flying and trains. Gas is expensive, tolls add up, and parking can be very expensive. One way to keep Budapest might be to fly Frankfurt to Budapest then train back to Frankfurt with stops in Bratislava, Vienna and Prague.

We have rented a car so no train or plane. We rented a car to make stops along the way between these cities but I didn't add that. So if we have more than a 2 hr drive anywhere, we are stopping in small towns to break up the drive. I'm still thinking we should cut out Budapest, nice as it is. It is the southern most so it makes sense and it will break up the drive between Vienna and Prague. Anyone have any suggestions where to spend a few nights between Vienna and Prague?

Cesky Krumlov is a popular stop between Vienna and Prague.

Does your rental car company allow you to drive your car into Czech Republic and Hungary?

I think skipping Budapest is wise and I vote for Cesky Krumlov too.