We had a similar (though admittedly not as bad) situation a couple of years ago. We booked a trip for late August to try to avoid crowds as much as possible. Then the school calendar came out and they had scheduled faculty in-service days earlier than ever before in district history. We were coming home on the first day DH was supposed to report for meetings - yikes! He immediately (this was probably February-ish) spoke to his department chair, explained that we had already booked and paid for flights and accommodations based on the dates of in-service historically going back forever & that we actually thought he would be back with about 3 days' leeway, he never takes his sick days or personal days, etc. The DC understood and it was approved. In your situation, I wonder if a similar appeal to your DH's direct supervisor to intervene with HR on his behalf would help? Supervisor could vouch for DH's responsibilities being covered while he's away (probably their primary concern), the amount of vacation time he has forfeited over the years, his tremendous dependability, his willingness to cover for others and come in during crisis situations for the good of the company, etc.
Or, I had these thoughts that are definitely not as good but might still make things workable:
(1) Could he agree to check in with the office on some periodic basis (a couple of times per week?) at scheduled times (so you could plan around, and so his colleagues would be available) to discuss any problems or current projects, etc.?
(2) Is there any work he could do from remote while you are away? Take 4 half-days for work so that everything back home keeps chugging along while he's gone? You & your son can shop (I recall he loves shopping!) or swim while DH works? Nothing TOO exciting so he doesn't feel like he's missing out as much! Small benefit here is that he could actually take 2 less vacation days that way.
(3) This seems crazy, but could he fly home for a couple of days in the middle of your trip to put in at work? It might cost as much for one RT flight in the middle as it would be to change 4 airfares, and if it keeps the rest of the trip intact, maybe it would be worth it?
Sending hugs and optimism your way.

There's just GOT to be a way to make this work. There seems to be plenty of time to plan for his time away in terms of doing anything possible ahead, ensuring coverage while he is away, etc. What would they do if (heaven forbid) he had some sort of medical issue and needed to be on-leave for a couple of weeks!?