I flew at 7 weeks with one of mine, out of necessity though, not for a holiday. Passport was through easily by then, I had a section and honestly it was a good recovery, no complications, but I had to be out and about registering baby, sorting passports etc MUCH sooner than I felt ready for. Given a choice I'd have waited a week or so longer to feel more human before even contemplating those tasks.
We moved the first immunizations forwards, so that they were had before we flew, but I don't know how early they'd let you move them. We got them at just over 6 weeks and flew 5 days later. I was adamant I wouldn't take an unvaccinated baby on a plane, given how planes are a nightmare for catching things anyway. Not sure the first imms would have given a huge amount of protection against lots of things, but it was at least something.
We were heading somewhere hot, and didn't venture near a pool for weeks, as pools were in direct sunlight mostly. The pool where we lived was heated, and was still too cold for a tiny new born. Weather-wise it was 32deg ish most days, and the baby was fine in the heat, no fiddly babygros to contend with (!), but we did stay indoors most of the time due to the intensity of the sun.
The plane travel was a doddle, as newborns tend to just sleep, so the crying babies (or at least, MY crying babies!) are usually 4 months or older in my experience. Under 4 months all of ours just slept through the flights...better than they ever did at home...
Our baby we flew with at 7 weeks had feeding issues, which we weren't able to sort out between diagnosis and travelling, but we figured it out out of necessity. Had I had more time, I'd have stayed in the UK to resolve things. I think it would have been better for both me and the baby. But as you're only away for a holiday it might not be such a big issue. Though friends that have had severe feeding issues have still been needing a tonne of support and equipment at the 6 week stage, so if you end up like that it could be tricky to take everything you need to ensure hydration.
There are a lot of good baby sun products out there, but I would absolutely double up or triple up on sun protection - creams, clothing, shade...
I'd say it is doable but not desirable. Even if all goes pretty well, like it did for me, I was not in the frame of mind to be travelling and holidaying. I just wanted my own bed and all the equipment and routines I was getting used to!
Good luck to you though :) enjoy it, if you manage to get there!