Not to rain on your parade but I would maybe consider doing a different walk than the Lecale Way. Its got great parts to it, no doubt, but also has shed loads of road to be walked (as do quite considerable parts of the Ulster Way). Parts of the coastal paths have became quite overgrown and a bit of battling was needed during the summer, might have died down a bit by now though.
If you go forward with walking it you could do so picking up food on your way, and also make it into a bit of a staggering pub walk. For water get it refilled in whatever pub is closest, I always had a pint in any establishment I asked for water in, for courtesy's sake

I would recommend dropping in at the Minerstown Tavern, me and Alan dropped in while walking the Ulster Way and got a great reception. It looked closed but there were a couple of peeps in and we ended up getting a couple pints donated to the cause and also the piss taken out of us

One of the guys in the pub even let us two stinky walkers sleep in his caravan and sorted us with breakfast as well! Also witnessed "skittles" being played something I'm sure quite a few people ain't seen or heard of before!
For camping from St Johns point round to Killough falls in around the right millage and offers plenty of opportunities to get a tent up and probably a bit of sea fishing or maybe swimming if you are a bit crazy in the head.
If I were to spend 2 days walking and wanted to do circa 40 miles I'd look at the Moyle Way, starting at Glenariff forest crosing Trostan and then through various forests, moorland and bogs before finishing in Ballycastle for the night. Its a nice enough section and probably the most consistently wild/off road part of the Ulster Way.
Finding your way through the forest before coming to Trostan is quite "fun" as the markers are tacked onto the actual trees, the fact that I found that entertaining might have something to do with having spent too much time in my own company. I ended up camping just down from Trostan in the worst rain I saw during the walk on the worst pitch of the walk in the least mountain worthy tent I have slept in, fun times.
From Ballycastle The Giants Causeway is 18 and a half miles following mainly the coastal path but also at the beginning quite a busy B road. The coastal path is definately one of the best sections of the Ulster Way and highly recommended for anyone who hasn't done it. If you wanted to hit your 20 miles then another couple miles would take you into Portballintrae or Bushmills.
That would be my suggestion anyways, even if it means you can only frequent a pub at the very end of each day, which is probably not too bad a thing.
*edited to remove profanity, thought it was a bit much