Definitely include Troon if you're going to Ayr to play Turnberry, Prestwick & Western Gailes. If you're going to East Lothian to play North Berwick and looking for something off the beaten path try Whitekirk or Luffness New. I would agree that Gullane is also a must.
In Fife, Kingsbarns, Crail, Scotscraig, Leven or Lundin Links, Panmure and Monifieth are possibilities for a second round after playing Old Course. Carnoustie is a must.
In the Highlands, Moray Golf Club, Fortrose & Rosemarkie and Boat of Garten are others if you're looking for a second round after playing Nairn or Dornoch.
Due to the time of year you're going, daylight last till 11:00 pm