It seems that the $SU(2)_1$ and $SO(3)_1$ Wess-Zumino-Witten models are quite different despite the Lie algebras being identical. The $SO(3)_1$ model has central charge 3/2 and is equivalent to 3 free Majorana fermions. The $SU(2)_1$ model has central charge 1, and can be expressed in terms of a compactified free boson (see for instance section 15.6 in Di Francesco et al's CFT textbook).
So unless I'm misunderstanding something, through ordinary bosonization the $SU(2)_1$ model should be equivalent to 2 Majorana fermions and thus equivalent to the $SO(2)_1$ model rather than $SO(3)_1$.
This situation seems very strange to me. Can someone point out where the global difference between $SO(3)$ and $SU(2)$ leads to a loss of a Majorana fermion?
Note that in this related question the brief answers claim the $SU(2)_1$ and $SO(2)_1$ WZW models are not equivalent, but frankly I don't see why that is the case. So perhaps my confusion with that question is related to this one.