Solution:

Let $m$ denote the minimum weight of a matching in $G$. Let $G''$ be the (multi)graph formed by taking the union of all minimum weight perfect matchings, but keeping edges multiple times. Note that $G''$ is regular. Now, assume that some perfect matching in $G'$ (equivalently $G''$) has weight $M > m$. Delete this matching from $G''$, and call the resulting graph $H$. $H$ is still regular, so it can be decomposed into a union of perfect matchings. Now using pigeonhole directly gives us a contradiction.

To show that all regular bipartite graphs can be decomposed into a union of perfect matchings, use Hall's marriage lemma to take out one perfect matching and use induction.