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Since the case of interest is $W_2(k)$ with perfect $k$ of characteristic $p > 2$, the answer is given by Ioan Berbec's 2009 paper "Group schemes over artinian rings and applications. In that paper (esp. section 3) he defines an essentially surjective additive functor to a certain semi-linear algebra category and proves that it is full (i.e., surjective on Hom's) and that the isomorphism property can be read off on either side. The same method of proof for the isomorphism property (which amounts to passing to a computation on the special fiber, where classical Dieudonne theory applies) works just as well for closed immersions and quotient maps. Thus, it is a simple exercise to deduce that the induced homomorphism between Ext-groups is an isomorphism. So that answers the question in the cases of interest. For more general bases things will be tougher; already for $W_n(k)$ with $n > 2$ I don't know a proved result which is suitable for doing hands-on Ext computations (but maybe Berbec's result can be generalized a bit, possibly imposing a "truncated Barsotti-Tate" condition).