Generally the second week of October is best which is why the wonderful Celtic Colours Festival is held at this time, this year from October 5-13. https://celtic-colours.com/
If you are interested in the music see if you can get tickets somewhere. This is a very popular time to be in Cape Breton so your first priority will be to find places to stay.
Here is a link to more information:
http://bit.ly/2ltXQbm Planning your trip
http://bit.ly/2mfOrsd Trip Reports
http://bit.ly/2nKbbyK Cape Breton
Let us know if you have any more questions.
Thank you for your advice and links.