The irritating smugness of SWISS comes from Switzerland's position in regards to the EU, placing it half-in and half-out of EU261and and its interpretations.
Fortunately for the OP, this journey was between France and Brasil, putting the carrier involved (be it based in Zurich or Timbuctu) firmly within the scope of the legislation and ECJ rulings.
Dealing with a claims agency creates very little hassle, indeed it removes most of the hassle involved in pursuing a claim in return for a significant chunk of any compensation received.
I'd certainly press SWISS further. A brief, formal letter giving them fifteen days to respond before you take action just might help them think beyond the routine denial of responsibility they have sent you. But probably not.
Don't muddy the waters with talk about hotels and prams. Focus on the issue at hand.
At the same time, approach one or more of the agencies. Most will tell you whether or not they are willing to take on your case. Given that they only make money if they win, you can take a positive response as encouraging.