On the other hand, feminist relationships seem to be happier.

If both results hold, it could plausibly be that feminism and unhappiness are both caused by something else at the country-wide level (the collapse of traditional meaning and values, perhaps,) or that feminism improves intimate relationships while worsening other aspects of life (by increasing labor supply, perhaps,) or that feminists make things worse for non-feminists (by not laughing at their rape jokes, perhaps.)

One interpretation is that we're in some kind of awkward middle stage where women are trying to fill multiple roles (perform well at paid job, look hot, take care of the house, raise kids, manage family's health, have a social life, etc.) I know I was raised with the message that it was important for me to be good at all of these things, and I would feel more a failure if I didn't do them all than I think my husband or most men would.

I can imagine that women's satisfaction would increase if role expectations were more egalitarian.

Even the paper you linked to doesn't claim that equality CAUSES unhappiness. A correlation might exist, but some of the authors' suggestions include:

if happiness is assessed relative to outcomes for one’s reference group, then greater equality may have led more women to compare their outcomes to those of the men around them. In turn, women might find their relative position lower than when their reference group included only women.

or

simply a change in how women answer the question.

(i.e. it's more socially acceptable to claim unhappiness)

I was told the exact opposite of that in my Positive Psych class. I remember multiple sources being cited but the only one I could find in the time I'm willing to devote to this conversation was a 2006 gallop poll thats mentioned in "Happiness" by Diener and Biswas-Diener. Anyways the countries that scored highest on happiness tended to also be high in gender equality.