You can SR for all 3 birds.Oh, I didn't know that they were available before you beat the E4. After you beat them, does he re-release them an unlimited number of tries whenever you speak to him, or only once, meaning that you'd basically get to SR for 2 of the 3 birds (1 pre E4, 1 post E4)?
Release them, RNG for 1st bird
Beat e4 round 2, release them again, RNG for 2nd bird
Beat e4 again, release them again, RNG for last bird