We have tons of carp here in the Great Sacandaga Lake and I see them spawning in the shallows of an inlet near my house, If the difference between koi and Carp is nil then there seems like there would be a good chance they would survive in a cool water pond but maybe not spawn.
If Brkt will survive in your pond they would be less likely to decimate small Koi but Brnt and especially tigers would murder your Koi. Then as mentioned would Koi roil up the pond to the detriment of the trout.