Sir Charles:
Yes, have also used that method, but with little success - The problem is that, with a lever-action rifle being fired rapidly in the "heat" of a scenario, one generally doesn't notice the case separation until you've tried to ram another loaded cartridge into the chamber - which definitely does tend to jam the shell fragment in quite firmly!
At any rate, as mentioned just above - even if a tight borebrush
will push out the broken case, that method is of little use to save the situation when the problem occurs in the middle of a shooting stage ....
That is why I am so interested in getting a broken shell extractor for .45 Colt - a nifty little gizmo which can be carried in a vest or shirt pocket and, if the need arises, can be quickly inserted partway into the chamber, whereupon you just cycle the action to extract the cartridge remnant. Having now seen one in operation (as stated in the other thread I started - in the Coffee Cup, I think) I couldn't believe how well it worked, or how I've done without this long ....
Which brings me back to the original problem - the only place I could find online who even list a .45 Colt version is Brownells, but they have discontinued it!