Spend a 3rd night in Madrid or Barcelona?

Barcelona. Not a close decision for me at all.

Ditto. Unless you love art (and you don't) extra time in Madrid will do nothing for you. We liked Madrid but mostly for the art and some good dining. If you do wind up in Madrid because of flight arrangements use the extra time to visit Toledo.

My vote is for Barcelona. I loved the relaxed vibe of the city, and the Gaudi architecture was fascinating.

And even though you're not big fans of art, give the Prado a couple of hours when you're in Madrid. I'm not a major art fan either--I'd much rather be outside wandering around--but I thoroughly enjoyed my visit to the Prado.

Definitely choose Barcelona.

One long day (on foot) knocks off Madrid's main 'non-arty' sights, possibly even with time for a Prado visit.
10:00 Royal Palace (opens 10am)
13:00 Plaza Mayor (lunch)
14:30 Puerta del Sol
15:00 Gran Via (Metropolis)
15:20 Plaza Cibeles
15:40 Puerta del Alcala
16:00 Retiro Park
17:00 Prado (closes 20:00 - ex Sun)

I know this is not answering your question, but...

If I only had 5 nights in Spain, I'd choose Barcelona or Madrid, but not both.

Keep in mind that 2 nights in any one place translates to one full day to explore, 3 nights = 2 full days.

Having said this, I definitely think Barcelona or Madrid deserve more than one day to explore.

If I had to choose between Madrid or Barcelona, I'd choose Madrid, but would spend my whole time in Toledo, not in Madrid.

Where are you going after visiting Barcelona or Madrid?

Most definitely Barcelona

What Priscilla said . . . ☺

For me, the main reason to go to Madrid is to see the incredibly rich and varied collections of oil paintings at its three great museums. Since you don't sound gungho on that, either stay in Barcelona or change trains when you get to Madrid and continue on to Toledo, plenty of charm and history there and very different from Barcelona.

I seem to be a minority but I spent a week in Madrid and felt that wasn't long enough.