It is possible, to prove Schanuel's lemma using the spectral sequence of a double complex. However, that does not really shorten the proof.
But I am interested in the generalization of Schanuel's lemma for projective solutions of arbitrary length (Ex. 3.15 in J. Rotman's Introduction to Homological Algebra). Is there a convenient proof of this using the spectral sequence of a slickly chosen double compex?