Let $V$ be a (real) Hermitian vector bundle over a smooth manifold $X$ of rank $n$. A $spin^c$-structure on $V$ is a principal $spin^c(n)$-bundle $P$ over $X$, together with an isomorphism $V\cong P\times_{spin^c(n)}\mathbb{R}^n$, where $spin^c(n)$ acts on $\mathbb{R}^n$ by special orthogonal transformations.
It is stated in several places in the literature that if $0\rightarrow V \rightarrow V' \rightarrow V'' \rightarrow 0$ is an exact sequence of vector bundles, then a $spin^c$ structure on any two uniquely determines one on the third. In Spin Geometry by Michelson and Lawson they give a proof for $spin$-structures using Stiefel-Whitney Classes, but I am curious as to whether or not it can be proved without this. For instance, one can describe a $spin^c$-structure for $V$ in terms of a bundle of irreducible modules for the Clifford algebras of the fibres of $V$ and use this to prove that if $V$ and $W$ have a $spin^c$-structure, then so does $V\oplus W$.
Any help or reference suggestion is highly appreciated.