The carex will be evergreen, the hakone not. Either than that, both will work equally as well. Personally, I like color variations of the 'Aureola' and think it would enhance the bark coloring of the birches very nicely. Don't care much for the 'All Gold.

I'm CRAZY for All Gold!

One thing to check out is speed of spread for the three plants. I don't grow either of the Carex you are interested in, but some Carex clumps or spreads slowly and some spreads fairly quickly. In my garden, Hakonechloa 'All Gold' makes a very slowly expanding clump so that it took probably 5 years to get from a one gallon pot to a two foot diameter clump. I like the bright gold to chartreuse range of colors it develops depending on the amount of sun it gets, and in my garden it is easy care.

In my experience, birches have shallow root systems that I think would leave Hakonechloas very unhappy. I'd go for Geranium macrorhizom(sp?)--THE plant for dry shade. It's semi evergreen in my zone.