Hi! You certainly can do all of Ring Road (Road 1.) Remember though that even this doesn’t cover all of Iceland!
I think the question is how many hours you want to drive in a day, vs. how many times you want to spend just a single night in one place. Let’s say it takes about 16 hours to drive all of Ring Road. Add in the trip to and from the airport, as well as time to drive to and from the Golden Circle and other attractions, and you’re at maybe 24 hours of driving. (This is all very rough, of course.)
So if you stayed in a different place every night, you’re averaging a little under 2 hours of driving a day. If you want to stay more than one night in one place, you’re looking at 3-4 hours of driving a day. If your 2-year-old will be happy to be in the car looking at the beautiful scenery, this might be okay?
But my advice is almost always to do less. You can’t cover it all, so cover less and explore more.
Oh, and the weather should be fine in August and September. It will be chilly, though– bring layers, maybe some wool, and good gloves and hats for everyone! (See our post about clothing in Iceland.)