I was also thinking about purchasing a handgun, but I was curious about how much they charge for: background checks, and is the tax Hawaii GET or Mainland where they are located?
Davidson's doesn't do any background checks, so there's no added charges from them. In the State of Hawaii, HPD does all background checks.
I haven't brought in anything from Davidson's in years, so their policy might have changed, but...
If you buy directly from Gallery of Guns, I don't think you will have any sales tax. The only tax you may see may be from your FFLs transfer fee.
If you buy from a local dealer who gets the gun from Davidson's, you may pay Hawaii sales tax on the entire amount.