I'm going to quote a comment that I think efficiently lays out some of the presuppositions that this question is based on, in order to express some disagreement with those presuppositions:
in principle one of these two options should be grammatically correct (albeit awkward), right? Which one?
It is not acceptable to add the -'s genitive (or "Saxon ...
The sentence is perfectly grammatical as it is. Here a friend of mine's is called a phrasal genitive. We say that mine's has double case-marking, an inner case and an outer case. See especially the discussion following the examples in , below.
Here is the relevant section of CGEL (pp. 479-480):
16.6 Head and phrasal genitives
The use of the Saxon genitive with the head noun dropped, to describe locations, is quite common in English.
We’ve run out of bread. I’m off to the baker’s.
Go to the doctor’s if you’re not feeling well.
He has been to London quite a few times, but has never actually seen St Paul’s.
She stayed at her uncle’s for over a month.
I left my trainers ...