When I moved to Hawaii for a few years, I shipped my car via Pasha. I think they have the only roll-on roll-off (RO-RO) service from the mainland to Hawaii. Cost for a car is around $1500 one-way, not including truck transport from wherever you or to the ship in Los Angeles. For a truck+camper, their website says to call with dimensions to see if they can take it.
https://www.pashahawaii.com/tools/vehicles/ship-a-car-to-and-from-hawaiiI was very pleased with their service, I dropped the car off at an auto lot near home, then about 10 days later they dropped it off at my house in Hawaii.
Another option would be Matson shipping, I don't have any experience with their car service (but I did ship a couple pallets of household items - they were good except for when the guy brought the pallets and tippped it off the lift gate when it was lowered, dealing with their insurance was a nightmare, I finally gave up and replaced a few hundred dollars of broken items at my own expense)
For interisland transport, I think you'd have to contract with Young Brothers for barge shipping. I've never used them. A while back there used to be a car ferry service between some of the islands, but that fizzled out. I actually had tickets for the ferry, but they shut down before my trip.
To be honest, I'm not sure how useful a camper would be in Hawaii, you can't sleep overnight in most public parking areas and it's not like there are a lot of RV campgrounds.