It's a small island; even if you ride every single track available you'll run out in a couple days. So I don't think there's much point worrying about optimizing your bike for the roads; you'll have plenty of time to get where you're going even if you walk the bike a lot. MTB is fine, the 47mm-tire gravel bike I rode was fine, the cheap hybrid bike my GF rode on 1.5" tires was fine. I don't see much point changing out equipment just for this. Trek 520 on 38mms will be fine.
The issues are: reserving campsites well in advance (unlikely you can just show up and wing it), considerable heat and lack of shade in summer, steep inclines on a few of the main roads, and most of all the very very fine powder moondust that will be all over everything in the dry season. I had to stop and scrape dust out of the front derailleur limit screws, as they packed up so hard with dust I couldn't shift any more. It's hard on chain lube too. Rainy season could be different yeah - but that's less and less of the year as time goes on.