Lot's a great suggestions here. Since your schedule is open to late April, May or June and you're flying in and renting for 10 days and want nice weather with spectacular scenery and you've got FF miles, why not Alaska?
Fly into Anchorage mid to end of May or June, pick up an RV and head down to the Kenai. The weather will be nice then, few bugs out, most of the tourists won't have arrived yet and there are dozens of lakes with camping. There are a number of developed RV parks and State parks in the small towns along the hiway. Also camping right in town on the bay in Seward or Homer with fishing, sightseeing, tour boats, etc.