I used to live in Appleton, WI which is about an hour south of door county. I agree with everything ode has said.
Horseshoe Bay is very nice from what I've heard and seen driving by the course. As stated is private but may be worth checking into to see if you can work something out.
Peninsula State Park course is very nice for a muni, but as mentioned can be very busy depending upon what time of year. It is the busiest between memorial and labor day.
Have played Idlewild and it was a nice course. Some water but not overly hard. The best thing about it, it wasn't as busy as Peninsula.
Hope this helps! Enjoy Door county. Some very scenic little towns on both sides of the peninsula.