Bart,
I live in Fort Myers. Starting in November it gets awfully difficult to find a place to play. Most of the courses that allow outside or reciprocal play during the Summer only allow members or their guests starting November 1.
In Lee County, where Fort Myers is located, the highest quality course with public access would be Old Corkscrew (Nicklaus). It's tough, but it is located in a great setting, is one of the best designed courses in SWFL and is the only course without housing in Lee County. River Hall (DLIII) is a great bang for the buck course. Plantation (Hurzdan/Frye) is good and in the past has allowed some public play during season. I looked on GolfNow and of those courses I would recommend the Verandah - Whispering Oak (Nicklaus) course. Eastwood (Devlin/Von Hagge) is an enjoyable municipal course that was a Golf Digest top 100 public course in the 1980's and Fort Myers Country Club is a Donald Ross course. Of those two, Eastwood is better maintained and a higher quality track, but I would rather drink a couple beers after the round at FMCC. If your client tries to get on either muni he needs to start calling at 8A.M. the morning before.
Collier County, where Naples is located, is probably more limited during season. I would recommend Bonita Bay East - Sabal (Fazio) and Twin Eagles - Eagle (Newly redone by Steve Smyers). I have heard good things about La Playa Resort's course, but I think you need to be a guest. Tiburon is way overpriced unless he stays at the Ritz or gets a deal.
Let your client know that Streamsong is only 100 miles from Fort Myers.
Good luck.