Thank you so much for your reply. I'm quite late in getting back to this thread!
The cats would definitely be flying in separate carriers, so no worries about that.
As for flying via Bilbao then driving, that was the "original" plan, but we found out pets can only enter Spain (internationally) through Madrid or Barcelona. Iberia is really the only option from there to San Sebastian.
The Lufthansa rep told us that so long as we had a two-hour minimum layover we could bring the cats to the pet lounge even if they flew in-cabin. Customs would actually be in Madrid, not Germany (as per the rep).
I looked at Condor but they don't allow pets in the Business Class cabin. If there is another airline that offers pets in International Business or First for the US to EU part of our journey I'd love to know. Once we get to the EU, economy is fine.