Lots of questions here. I'll try to help.
1st. Did you build this house from the ground up? Foundation and all? It's a concrete slab, yes? If yes to these questions, then common sense dictates that a 6 to 10 mil polyethylene vapor barrier be installed UNDER the concrete and over a sand or gravel base, to help prevent the concrete from naturally wicking up ground water. THis MAY be the problem. I don't know. You say you put down plastic OVER the slab, then put down 3/4" T&G backer board? What the hell is that? NOFMA and NWFA specs call for 3/4" CDX plywood as an underlayment for nailing. How was the "backer board" installed? Nailed? Screwed? Glued? Tapcons? What? Also, are you sure the wood is truly AMERICAN Mesquite? American mesquite is a VERY stable wood, one of the best for stability. However, buying from LL is suspect as it MAY be some weird asian import called "mesquite" when in fact, it is far removed from true mesquite botanically. You need to check that first, IMO. Got to know EXACTLY what you are dealing with! Another thing to check is RELATIVE HUMIDITY. American Mesquite is from TEXAS and you are in FLORIDA. Quite a swing in humidity ranges. That flooring was probably milled and dried for the local market and you aren't it so it may be that it was not properly acclimated to the conditions it was to be installed in. Also, is this floor a solid or an engineered, multiple ply floor? If it's a solid, and I suspect it is, they are typically nailed down. How was your floor installed? I like the part where you mention you hired a "English" carpenter to install the flooring, like they have something over us Americans. I can tell you, I've forgotten more about flooring than ANY "English" carpenter ever knew. AS to how to fix it, let it sit for 6 months and see what happens. If if doesn't get better, then you'll need to rip it out and get some QUALITY flooring from a reputable distributor and hire a real hardwood flooring contractor to install it.
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I did this thing by the book and took every precaution just to avoid a call back.
No you didn't. You don't even own "the book"so how can you say you did it by the book. I, and many other NWFA members, own "the book", called the NWFA Technical Reference Manual and if you had done it by the book, you would not be experiencing the problems you are facing now. OK, lecture over. Sorry, but lots of folks come here and tell us pros how "correct" their installation was and what's wrong with the wood they bought at a discount somewhere. Most of the time, and the inspectors will back this up, it is installer error.