I've never swept a chimney, so this is just thinking out loud based on nothing.
If you know how many rods it takes to reach the top from previous experience, can you not just sweep carefully to the top and stop when the resistance changes?
I would suggest fitting a suitable cowl with some kind of mechanical fixing so that a gentle prod with a brush won't send it flying, and just go easy when you know you are nearing the top when sweeping. Some of the cowls look to have quite substantial strap fittings or whatever and would be quite difficult to dislodge.
At least then you know you only have a one off expense fitting the cowl, you've rid yourself of nesting crows, and you can still give it an occasional sweep when necessary. Best of luck.
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