Thank you to everyone for all the replies! Only a few days to the trip now...
Based on tripadvisor the birthday dinner will be at Fiskfelagid, although I am a bit worried that it's not mentioned here at all. Bad choice?
We have 4 other nights as well, so 3 Frakkar sounds definitely like something for one of them. I've also stumbled upon Dill in my investigations, but it is quite pricey... Let's see.
One thing I forgot to mention was that my girlfriend needs gluten-free food. Do you expect that to be an issue, both when it comes to availability and the attitude towards such requests?
I will make sure to share my experiences here!