GAW.
Rav Moshe Z’l writes in a teshuva that one need not give his wife a get if she wont cover her hair. However, nowhere in the teshuva does he posit it is because of the importance of shalom bayis. I wish I could cite the teshuva or at the very least where it can be found. Hopefully someone else will find it sooner than me (I have no intention to look for it until sunday at the earliest).
Also, your assumption(?) or is it assertion from the sotah is way off base. The gemara is clear about the importance of shalom between a husband and wife, and it cites the fact that hashem allows his name to be erased, not the fact that he permits pritzus in the beis hamikdash. May I suggest Rashi on the passuk in chumash who explains why we do that. Also see how he translates the word “uncover”.
As for what hashem deems more important, I wont be so presumptious to make such a claim.