Just got back two weeks ago....glad everyone else here is doing so well, but we were on roughly the same budget. Stayed at the Hotel Prince, just around the corner from the Rue Cler...it's mentioned in Rick's book...perhaps a little rough around the edges for the more well-heeled, but we found it clean and comfortable...and best of all a short walk to the Eiffel Tower, Napoleon's Tomb, and the Seine....we ended up walking much further, but we like walking, and with record floods, the buildup to the Eurocup, (IMO completely justified) Labor Strikes, and the President of South Korea in town there was lots to see. I just punched the 2nd week of August into booking.com, they show a little bit over your budget at $320 for 3 nights...you're in peak season, we just missed it...but Rick says book with them direct to get a better price, and supposedly they give a discount to folks who flash their RickBook, though they don't if you book on-line. There's also a grocery store around the corner, you can save tons of dough by eating like the French...a Baguette, some nice cheese, some fresh fruit, and a little cured meat...and a REALLY great bottle of wine for less than $5.
We took the RER and Metro to the hotel from CDG (very inexpensive), but on the way back strikes and a flooded Seine made that less than reliable, the hotel desk recommended a shuttle bus....36 Euros for two of us, plus tip...I gave 5...one of the few places in Paris where they expect them.
CDG is really quite far from the center of things, don't think it'd be much fun to stay there...and FWIW our last 4 nights in Paris were the last of a 2 week trip, we started with a rental car at CDG and drove to Reims-Bastogne-Bruges-Rouen-then camped out at Omaha beach in Normandy-then Chartres and back to CDG to drop off the car...actually enjoyed that part more than Paris. Have a great trip!