Dolores, I am wondering if the palaces your husband is referring to in Bordeaux are the castles of the Dordogne? Or maybe some of the chateaux associated with wineries? Does he have something specific in mind?
Tocard's comments are spot on regarding flights (open jaw would be better...that is not two one-way trips), and Nice-Bordeaux-Paris is a good, logical route. Book your flight to Nice (plane change in Paris...very doable). I also would rather have one long train day from Nice to Bordeaux than your first choice. Or even a flight from Nice to Bordeaux. There is a TGV from Bordeaux to Paris that takes less than three hours, so definitely choose that for your return to Paris. We have taken that and it is easy and fast.
So, here's what I see
Day 1 arrive Nice, stay 4 nights (3 full days- days 2, 3, 4)
Day 5 train or fly to Bordeaux, stay 4 nights (3 full days- days 6, 7, 8) (travel takes up lots of day 5, depending on your mode of travel to Bordeaux)
Day 9 TGV to Paris, stay 5-6 nights (4-5 full days-days 10, 11, 12, 13 and half-day on day 9). (TGV travel gives you an extra half-day in Paris)
I assume you have 14 days. If departure is on day 14, then 5 nights Paris. Day 1 is kind of taken up with arrival, jet lag, etc. Day 14 is shot for sightseeing as you will be heading out to CDG and flight home. You have 10-11 full days of sightseeing/activities.
All very doable and not rushed, with just one long travel day.
BTW we took our two when they were 10 and 8 and they enjoyed everything! We didn't have to focus on kid-type things either. You will be surprised at what they will appreciate and enjoy!
Enjoy yourselves!