However, once they consent with you, you can get whatever resource you want.
Before talking with big boss, all the dusts should have been settled. Come to big bosses with solutions/choices, not problems and headaches (in this case, he will have to deal with this sr. director)
I have been leader to many projects, and I would love to have any sr. management guy to "join" my team. I will use his name as much as I can and get job done much easier. (you can always say that XXX agreed!!) but most importantly, get the job done so those directors will feel happy to have supported you.