Your Iberia/AA flight should be bookable with Iberia Avios, but it doesn't show online and probably wouldn't make sense to do anyway. I do see it on ba.com for 60k + $2xx.xx per person.
Your situation illustrates why frequent flyer miles are rarely a good use for international economy. You have the CSR and (soon-to-be) 240k points. Cash flights are under $1,200 per person. So use ~80k Chase points each and book a nice 1-stop each way itinerary. Even if you could find, say, round trip using United miles, you'd use 85k miles + ~$125 in taxes/fees + you have an 11-hour layover on the outbound you've found. Much better deal to spend 80k Chase points, have reasonable layovers, and even earn miles for flying.