I wouldn’t change from the camera you already have access to. Personally I feel like I get better photos out of a full-frame than an APS-C.
Regarding light, for eyes it’s often better to use continuous lighting rather than a flash as a flash, even offset, is uncomfortable for the subject. If you do use a flash, make sure it’s at the lowest setting.
There are a ton of videos and articles specifically about taking photos of people’s irises. Search out those and watch/read some of them.