It's definitely possible, but you might miss out on a lot of things because you won't have the time. That's a lot of driving each day (it always takes longer than you think, partially because there are so many great pulloffs).
I went in Late June/Early July so we had lots of daylight. You should still have plenty in August, though probably not the same eerie daylight all the time situation.
This is what we did. I hope this is helpful in planning. We just stuck to the south coast, and really maximized our sights there. Seems like you are going the exact same amount of time as us.
Day 1: Husband arrived early morning. I was coming from Europe from a conference so he had a half day to do stuff before I got there. He did the Reykjanes peninsula and loved it.
That night we drove 2.5 hours near Vik to an airbnb. We then went and did the Black Sand beach and Skogafoss from 9-11 pm (big advantage of the daylight). Stayed near Vik.
Day 2: Drove to the Glacier lagoon. From where we were staying it was about a 2.5 hour drive, but you'll want to stop much more often(lots of great pullouts with cool sights and information posts). That day we did both glacier lagoons, the diamond beach, and did two hikes in skaftafell. Left around 9 am and got back around 9 pm. Can't imagine fitting any more in that day. We got lucky and had beautiful weather. Stayed near Vik.
Day 3: Took the ferry to westman islands (highly recommend). Had bad weather but made the best of it. We climbed the volcano twice since the weather got better as the day went on. Stayed on Island.
Day 4: Ferry back to mainland. Saw Seljandsfoss, did the Hot Spring river hike (probably only thing I would have changed), and part of the golden circle. Stayed in Selfoss.
Day 5: Golden circle (did Laugartavn Fontana here instead of blue lagoon and would recommend). We did a lot of the less common stops here too. Stayed in Reykjavik
Day 6: Reykjavik
Day 7: Did Reykjanes peninsula again so I could see it, and my husband didn't mind redoing. Flight out around 4pm.
I totally understand wanting to do the full ring road, but I'd encourage you to stick to a smaller part, even though it means backtracking. You'll spend much less time driving each day and more time enjoying things. We didn't make it to the Snaefellsnes peninsula but have heard great things, so that could be an option as well. Also depends on how much hiking you want to do. If you aren't into that as much, then you can breeze through things quicker. But definitely take weather into consideration. And just enjoy things even with crummy weather!
No matter what you decide on for itinerary, you'll have a great time!