I have done axis on Lanai. There's basically 2 types of land - state and company. State land (CGMA) you need to apply for, company land you essentially pay a trespass fee to hunt on (it's managed by "the company", I think they're called Pulama Lanai or something). State land has seasons, archery is usually feb I think, and last year it was does only for archery. Company land I believe you can hunt every day, at least the archery sections of it, and you should be allowed to shoot bucks. You can also pay for an outfitter on Lanai, I think they have their own private section of company land just for them? I've never done it tho, it's pretty expensive.
If you got a few days, the state land hunt or company land hunt isn't too bad. Lots of deer everywhere. Don't know anything about rifle or muzzleloader hunting there, I assume the regs are different than archery but never done it. I do know that company land permits for rifle cost much more than archery, and may only last a day or so whereas archery permits last a whole year (at least they used to, not sure about this year if it changed or not).
Anyway, if you got questions about areas and such on Lanai feel free to PM.
Also, check out
https://www.lanai96763.com/information they have a hunting section if you scroll down with info on hunts and pricing and stuff.