Given how women initiate most divorces, it's probably best to set it up so they are the proposing side.
But this isn't raw Gale-Shapley-- the participants aren't ranking all hundreds/thousands of others. In particular, the preference matrix is based on identical mutual compatibility scores, so there's a unique stable solution (not multiple stable ones!), and there's no "side" that benefits, since the solution is identical no matter who does the "proposing".
> the preference matrix is based on identical mutual compatibility scores
Is that modeling the problem correctly? It seems pretty rare for two partners to have equal amounts of love for each other.
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If stats from dating websites are to be believed, all women tend to flock on all the same tiny percentage of men and ignore the rest. If women initiate, I’m afraid you will see only a few men getting any pairing. Men are likely to be less picky as we see in the wild.