YMMV on this one. Yes, they will send your bags all the way through. And yes, they will charge you. Generally, the policies of the originating flight take precedence, i.e. HA will charge you for bags, at least from Kona to Honolulu, at the inter-island rates. If you have elite status (i.e. free bag check) with Delta, then it is a matter of figuring whether it would cost more to just pay the mainland rates through HA, or splitting it up between HA and DL.

One thing I did (because I have PM status with DL) was not pay the bag rate online; I waited until I got to the counter to check the bags, and ensured that I was only paying the lower inter-island rate, since that was the only time my bags were on HA metal. Once they have your money, it's probably harder to get it back.