Whatever your opinions on the Institute of Texan Cultures might be (personally I'm a sucker for the space age Futurama designs), you can't deny it's content is a literal smorgasboard of history. It would be criminal to box it all away never to be seen again, but I sincerely doubt if there comes a time when the building is demolished for the eventual Wembydome would it ever re-open?
The ITC is not an actual legitimate museum, it is not accredited as such. The building lacks a proper loading dock, or adequate fire safety system, or security command center. It has asbestos. In that sense it's not really comparable to something like the Witte or the McNay, big established legitimate organizations that turn profit (I'm assuming, at least more than the ITC ever does/did). If it is to be closed it's unlikely it would ever re-open in the downtown or elsewhere.
Off the top of my head my proposal would be to relocate it's collections database and archives to appropriate adopters; San Antonio College, the Central Library, Texas A&M Archives are some possible candidates or UTSA would just keep them all anyway, and to then refurbish the original exhibit areas into public displays to be installed mostly in the Henry B. Gonzalez Convention Center (which I predict will be expanded on
again fairly soon), but these could also be offered for display in various tower lobbies (Weston Center, Frost Tower, the major hotels like Hyatt and Marriott). Texana-related items would have an obvious second home at the Alamo. At the very least moving the exhibits into the Convention Center assures it will forever be seen and enjoyed by the millions of tourists and locals alike who enter the space, but having them distributed as a kind of rotating art pieces across the downtown skyscrapers would keep their image around.
The only thing I wish that this future Wembydome has is it keeps the iconic Flags of All Nations (obviously not all of them, I think there's about 40 total) and walkup as you depart the ITC. Not sure what this could look like, but I hope that the image of the true flags of the nations beating and shaking against the wind is something kept as a callback to what once was.