You could probably 3d print a square or hex shaped thing that has a slot on one end to fit onto the stopper, which will give you a better angle to grip it, work it from side to side or twist it. Wood would be another easy material option to try to make a little tool out of.

That said if it’s really stuck enough to break the handle, then you’re going to need a new one anyway, and at that point destructive measures and eventual replacement may be your only option. As any mechanic with a blowtorch will tell you about a seized fastener, “it can’t be stuck if it’s liquid”