Advice for Late July-early August Trip

Consider carefully if you will be on your game for a nearly three-hour drive after an overnight flight (assuming you are flying in from US unless that is a missing piece of info).
Driving in and out of Athens is going to be hectic to say the least, so please share where you are staying and add if you have driven in a large unfamiliar city like this before.
Your day trips are pretty long--I know you are trying to work around some complications, but this is concerning to me. I think you would be better off with day tours. If you do that, then you could fly to an island after the competition. If you don;'t, then you could drive to the Peloponnese after the competition. You could spend those early days touring Athens and doing day trips.
Rental car on ferry is up to the car provider so you have to check with them.

I completely agree with valadelphia.

You'll be arriving, I presume, after a long transatlantic flight, and driving to Delphi in these conditions is not only unreasonable but also dangerous for you and others. Especially since it will be a Friday at the end of July, when Athenians leave Athens for their August vacation. No need to add your inexperience with driving in Greece to the traffic jams.

So, slow down. Stay in Athens when you arrive on Friday.

In Athens in August, it's possible, even probable, that temperatures will reach at least 40°C (105°F) or even 45°C, forcing archaeological sites to close from 11 a.m. to 5 p.m. for safety reasons. These days, the temperature is around 38°C during the day and doesn't drop below 29°C at night. So, even if you're open to activities, I guarantee you'll want to slow down.

If this is the sport I think it is (but I could be wrong), the events aren't taking place in the center of Athens, but a 30-minute drive from the center and far from any metro station or bus line that would connect to the city center. Have you planned how you'll get there?

Regarding the period from July 31st to August 5th, it's only 5 days.

Option (a): Peloponnese can be a good choice provided you don't want to visit the entire Peloponnese.

Option (b): Cyclades, 5 days is barely enough to visit one island. If you like crowds, choose Paros or Naxos. Otherwise, Sifnos can be less crowded. Also, see Serifos or Tinos.

Option c: The Ionian Islands will require you to take a flight, at least to Kefalonia since Lefkada doesn't have an airport. Is this really necessary with the limited time you have? In my opinion, no. These islands, at the western end, are almost closer to Italy than Greece.

We did, as you are considering. Drove directly to Delphi on arrival to Athens. Spent a lovely night there, visited the Delphi site in the morning and drove on to Meteora for two nights. Worked great for us. From there, drove to Pireaus and took the ferry to Sifnos. Loved Sifnos.

Have a great trip!