Lufthansa definitely allows pets in the cabin in all classes of service from Economy to First. Beware that LH is rather strict with enforcing their weight and carrier size regulations - you may not be able to put both cats in one carrier depending on their size.
The fact that they have a pet care facility at FRA is irrelevant for you if you take them in the cabin where your cats will stay with you at all times - their 'animal lounge' is designed for animals transported as cargo or checked baggage where they are in LH's care.

Generally, it's best to avoid multi-airline itineraries with pets. The golden rule for travel with animals is as few stops as possible, on as few airlines as possible.
Changing from LH to Iberia means your pets need to be booked separately on both airlines, and you will need to go landside at FRA (which means passing immigration and customs in Germany, so you will need to have the cats' EU paperwork prepared for Germany) to check them in and pay the fee again with Iberia even if your trip is on one ticket. Animals are never interlined between airlines even if your bags are.

I`d not do this but fly Lufthansa to Bilbao all the way (SEA-FRA-BIO; they don't fly to EAS unfortunately) and drive the 60 miles to San Sebastian from there, which takes about an hour. This way, you have a seamless flight with only one easy transfer and short 90-minute layover in FRA.

As you say that you would usually book economy, Condor has very attractively priced business class offers from Seattle to Europe via Frankfurt, with lie-flat seats and almost as many bells and whistles as you'd see on Lufthansa. If you don't mind having to change airlines in Frankfurt, you will save a lot of money by flying with them instead of Lufthansa, especially so if you are flying on one-way tickets. They do accept pets in the cabin in business class, with a 6kg weight limit per carrier.