That happens very frequently with all airlines - maybe 20% of my visits/transits to US. If you have other travel soon that will result in a passport stamp, then I'd be inclined to do nothing. Based on past experience you will be questioned at immigration next time you visit USA but as long as you can show you left the US quickly (by arrival stamp elsewhere) they can see you didn't overstay and everything is okay. The card not making the processing centre is an everyday occurrence and US immigration are well used to this scenario.

OTOH, if you wont have such proof that you didn't overstay in the US then I would send it to the processing centre by registered mail. The last thing you want on next visit to US is to be questioned for hours about your previous "overstay", with some risk of being denied entry.

Lastly, whatever you decide to do make sure you remove the stub from your passport. If US immigration see it there on re-entry they will ask a lot of questions.