Suggestions please - spare days in itinerary

Public transportation is not a problem at all. There's no train station anymore, but a bus center.
You can e.g. go from Miyazaki JR Station or Airport, but also from Kumamoto.

A day trip is possible.
However, one great thing about Takachiho are the kagura dance performances in the evening (until 9 p.m.). If you want to see that, I recommend staying for the night as the last bus leaves before that! You can then move on the next morning. :)

For me, Yanagawa was one of my better trips close to Fukuoka. However, it really depends on what you find interesting.

Yanagawa: https://www.japan-guide.com/community/mfedley/report-2676

If you have a wet day in Fukuoka, then there are some reasonable museums in Kokura: https://www.japan-guide.com/community/mfedley/report-2684

Yoshinogiri Park is quite nice - but I would dress warm when you visit and make sure there is little chance or rain.

https://www.japan-guide.com/community/mfedley/report-2692

Ohori Park in Fukuoka also has a reasonable smattering of gardens and museums as well: https://www.japan-guide.com/community/mfedley/report-2713

Thanks very much both of you!