A resounding NO. If you can get funded at Sussex by going with Media Studies, then stay there. In a PhD, it doesn't matter what your degree field says (well, with a few obvious exceptions...like....history vs. chemistry, or something). What matters is the research that you're doing, period. Your job prospects after the PhD are dependent primarily on the work you've accomplished and your publications, but it also REALLY helps to be well-connected: if Sussex is a more well-known institution in your field, then you may have better opportunities to network. And don't forget, being happy where you are (and your girlfriend being happy too) should not be neglected...these are hugely important considerations.
Can you get a compensation level similar to the German offer by going with Media Studies? Also, talk to your advisor regarding your situation, and discuss how it would impact your post-degree career prospects, he or she will probably have some great insight into this and might have some ideas on how to get funding at Sussex.