I reckon I could get that chain out without tools.

Chains are tricky buggers; there's a real technique to dealing with them when they get tangled. I wouldn't be surprised if that chain eventually comes out easier than you think...

I'd take a jockey wheel out of the rear derailleur or split the chain if it has a quick link first, just to make life easier. I'd put a glove on my right hand and pull on the lower length of chain (with the bike upside down, paying careful attention to the direction I'm pulling), while using my left hand to pull on the other end to get it out of that trap while a helper slowly turns the crank.

By the way, I'm enjoying the irony of those tangs causing the problem they're meant to prevent... maybe you could just bend them out of the way with some longnosed vicegrips.