I imagine they'd let you keep it onboard, though it might not be a simple yes right away.

I witnessed a guy in a similar situation recently, with medical equipment he needed to keep with him. The one big difference, however, was that his bag was not of legal size. And it wasn't just slightly off, it was so big that it literally could not even physically fit in a overhead bin. I still wonder how he got it through security to be honest. Long story short, he succeeded in his quest to keep it with him, after many supervisors, lots of arguing, and a lengthy delay. I'm not sure exactly where it ended up being stored for the flight, but it stayed onboard.

Personally I would've just kicked him and his luggage off, and not inconvenienced the other 150 people onboard that actually followed the rules. But I suppose that's where the fine line of customer service comes into play with a society that is inherently very egocentric.