Funny, I was looking into the Cape Lookout area. How is the weather around early April?
There's always some risk when camping on our barrier islands but odds are you'd see low 70's for day time highs with nights falling into the mid to lower 50's. We've tent camped there a few times in mid March without too much discomfort. Wind may be an issue so take stakes at least 15" long. Bugs will not be a problem. If you want to take a vehicle over, there are two choices for camping - Portsmouth Island (North Core Banks and the Village) or South Core Banks (Cape Lookout Lighthouse). You cannot drive between the two.
At this time, the public restrooms and showers on Portsmouth are not operable due to last year's storms and there is no potable water available on this island. I don't know when or even if those will be restored. I think these same facilites on South Core are functioning but a call to the Park Service should be made to determine current status. My preference is Portsmouth and we have a trip there planned for May regardless of the conditions. You will not find a more remote ocean side camping experience anywhere on the east coast.
I was part of a group that made a late March trip to Portsmouth a few years back and one member did a good three part trip report.
This is a link to the first section and a link is provided at the end to part 2.
Al
Edit to correct link