I like the idea of a split stay as well. Your family would enjoy it!
My DH had a conference at the Hilton for a week once in August. I was 3 months pregnant, and it was so HOT that it was really dangerous for me to be out in the heat. We stayed at Shades of Green. I enjoyed just going over to the Grand Floridian, eating in their snack bar, then getting some tea or water and reading a nice novel in a quiet area in the lobby. Got a haircut there too! Of course staying there was out of our budget that trip! Anyway, DH had to drive to the Hilton every day for his conference, then back in the afternoon when they were done. It wasn't a lot of driving, but I felt bad for him to have to do that every day.
I think if we had stayed at the Hilton, for at least part of the week, it would have been easier for him.
Maybe start the trip at a Disney resort (that way if there is a good package available, the family could really take advantage of it) then do the Hilton a few days later.
July 4th week - very busy at Disney I hear. Very hot, yes! Take that into account.
A good place to start is finding out, what do you have to pay for, and what is already paid for? If you have a per diem, and can use it anywhere, or are they willing to only pay for the Hilton (doubt it, but wondering).
You will need to take into account travel time to and from the Hilton, if that is where the meetings will be. Driving is the best option if you are at a different resort.
You may want to look into a close by condo, maybe a home with its own splash pool? I would only consider that, as you are looking at a lengthy stay. It would be a good homebase for the family - it will be hot so a pool would be nice. I would like a kitchen and laundry room personally, as eating out is fun, but two weeks is a lot! Sometimes you just want a salad! If you did that, you may want two rental cars - or the spouse could drop you off at the Hilton, then park at Downtown Disney, and get to the parks from there?
This sounds like it will be fun! Enjoy planning!
PS - I've never stayed at a house or condo offsite, but just thinking about it being two weeks.