Without sitting to verify the details in full, here is a sketch of a proof:

Consider Lauchli's construction of a vector space with two bases of different cardinality, as outlined in Jech **The Axiom of Choice** in problem 10.5 (p. 149). The construction is to create two infinite sets which span isomorphic vector spaces, but have different cardinalities.

Replace "countable" by $\aleph_2$ and "finite" by $\aleph_1$. The result will satisfy $\sf DC_{\aleph_1}$, and the conclusion *should* still hold. Moreover, you can take the free abelian group generated by those sets instead of a vector space, and it would solve the problem.

If one prefers to do that by forcing instead of atoms, then it is also possible, although slightly more technical.