HAZEL
(just in case edit below)
if BARGE-1, E must be 4th and L must be 5th.
BAGEL-3, BAKED-2. since A is proven at most 1 of B__E can be correct because both words have BA_E_ and BAKED is 2 while BAGEL is 3. If neither B---- nor ---E- were correct then both --G-- and ----L must be correct which leaves no possibilities for BAKED being 2 because B---- and ---E- are wrong and the other 2 slots are taken by --G-L. Thus exactly 1 of B--E- is correct. if BARGE-1, that 1 is taken by the A, disproving B, which means the 4th letter must be E.
If this is the case either --G-- or ----L must also be correct from BAGEL-3 (B misproven, A proven and E proven). From CAGEY-2, the G is eliminated because -A-E- are proven. This leaves only L.
Proving -A-EL