Do not, under any circumstances, even think about running a SaaS business from Europe. EU has made invoicing hell for small businesses, especially those selling SaaS. You will spend time on things like VAT number checking, verifying customer's country using their IP (yes, I know), several kinds of invoices, prices inclusive or exclusive of VAT, calculating VAT for each EU country differently (and in different currencies), reporting VAT, and lots of other silly annoyances.
And if you think this is a "solved problem using a third-party service", you haven't tried it yet.
The only disadvantage of being US-based is that the first serious lawsuit will basicaly crater your business. Otherwise, you are way ahead of your buddies in Europe (like me).