Nice idea for your trip ^ However, exceedingly late to get non stop business class for summer travel.

Two suggestions: 1) sign up for expertflyer to get alerts for availability that pops up and 2) If you find anything even close to your dates on AA or UA, book it. You can change dates free if flights for your exact itin open up for preferred dates.

Paris in early June was great last year. Started to get very crowded in the middle of the month. I think it was both because school was out and possibly a fashion week. My business class ticket was using AA miles on Air Tahiti Nui (CDG-LAX)