I can't point you in the direction of a good place to get it fixed but I can sympathise.

I am in the same boat. I bought a Prada when I was in the Greek Islands last year and this week the EXACT same thing happened!! The stitching has started to come undone on one of the straps and is dangerously close to breaking. I'll probably have to find somewhere to get it fixed the next time I want to use it - in the meantime I have my new Marc Jacobs to keep me company!!

Hope your baby is on the mend soon