What you need, mrodoc, is a chimney balloon.
I fitted one in our (unused) fireplace and it works a treat.
If fully inflated it makes a wonderful seal, though of course to prevent damp I don't fully inflate it, but I'm sure you get the idea..
They come in various sizes and are inflated with a flexible pipe so you don't even have to have your head in the passage (if ingress is an issue).
try
https://www.chimneyballoon.co.uk/
https://www.amazon.co.uk/Chimney-Balloon-Medium-Chimneys-60cmx30cm/dp/B009MYL5PO
etc.