A Rytter,
I would narrow your options down to 1) Newark (Northeast) 2) Chicago (O'Hare).
I personally wouldn't want to go much further south than D.C. in August as it'll be very hot almost anywhere you go in that direction. If you went south through the Carolinas (Pinehurst, Charleston) there is some great golf....but it'll be hard to play some afternoons due to heat and t-storms.
"Access" to the private clubs of the Northeast are tough tickets, and there aren't enough great public courses to warrent a special trip to the area from Europe...IMO.
Your best bet? Chicago (O'Hare). Tons of flights going in and out means quality flight time options and usually cheaper flights.
You could built an incredible itinerary of public/resort courses and accessible private clubs just by driving around the Great Lakes. Assume you fly into Chicago and rent a car. 1) Chicago is a world class city in itself, worth eating a day or two to check out, and is perfect in August (food, museums, theater, hot chicks on North Ave. beach
) and has a TON of quality golf with a fare share of local GCA guys that I'm sure would be happy to join you for a round or two. Driving northeast into Michigan you could hit up Harbor County (Dunes Club, Lost Dunes, Point 'O Woods, Harbor Shores) before heading further north towards Traverse City (Arcadia Bluffs, Kingsley Club, Greywalls) before rounding the UP and driving over to Duluth (Northland CC) then down to Minneapolis-St. Paul (tons of great golf, similar to Chicago with plenty of GCA guys...myself included) then a ~5 hour drive to Lawsonia Links (public) in Green Lake WI, then 45 min to Erin Hills (public), another ~45 min to Kohler (Whistling Straights, Blackwolf Run - both public) before driving back through Milwaukee to Chicago (~2 hour drive).
That's a whole lot of good to great golf in an area of the country that is fantastic in August weather wise. Sure, not every course on the list is public, but many would probably make an accommodation for an international traveler with a well-worded letter (unlike many parts of the Northeast).
Hope this helps. Feel free to PM me if you need more specific help in the region.