Griffin is the one i’ve thought about the most (also happens to be my favorite, if I could put griffin as a favorite taxon on my home page I would).
I think class-wise Aves would be best. It would be in its own order, family etc. Maybe something like Falcoleo equivoros (horse-eating falcon-lion) for a scientific name.
I am not sure how large the wing span must be for it to have flown. The bones would be mammalian in shape (except for wings and head) but pneumatized, maybe with a mineral that hardens the bones so stress does not crack them. The wing bones would be attached like a normal bird, maybe with a smaller keel then normal so the front limbs have a place to go. No idea about musculature, digestive system probably lion-like.
edited original post to include basic descriptions of each.
Also just remembered a fandom I saw once: https://fiction-taxonomy.fandom.com/wiki/Fiction_Taxonomy_Wiki. They have griffin in it’s own class, Mammaves, and a scientific name of Opinicus gryphus.