Nope, Tiamat made Red Dragons serve Githyanki because Tiamat wanted a deal with Disapater so Gith (the legendary hero) was imprisoned at Dis (that Hell-ish city) as a deal. Gith's powerful soul empowers Disapter.
Red Dragons don't get a say. If they tried to repel, Tiatmat has 100% ownership of Red Dragons can just destroy the rebels. Not like Bahamut is gonna advocate for them.