I think I may have found a clue to your problem on the main blog:
deb, by now you can tell ME how the clarified butter experiment worked. i tried it of course when working on the pastry bible but clarifying butter denatures it and it does not perform like lard. it is a totally different type of fat structure.
That was Rose, replying to a question about using clarified butter instead of lard in a pie crust. Link: http://www.realbakingwithrose.com/2005/11/introduction.html#comment-98134
So, when you browned and cooled the butter, you ended up with a fat with a different structure—even though the molecules were (mostly) the same they were arranged in a different way.
As to what to do about it…that could be tricky. Maybe you could brown only a portion of the butter? You might need to do some serious recipe adjusting and I’m not enough of an expert there to help you. Have you tried searching for some Brown Butter Cake recipes on the Web?
Good luck and happy baking!