I see that you're AA Plat, which means you'll have access to OW Lounges.

Not sure if this will work, but you could potentially see if you could leave your bags with an IB Lounge. (They have a few lounges in Term 4 at MAD.)

I've left my bags in a lounge before during stopovers, but I was also doing it as an AC member in a domestic AC...YMMV with partner lounges.