Kefalonia is a lovely Island – I took my Mountain Bike there in 2002 and had a great time.
You will probably be restricted to the unpaved dirt roads (that are reasonably well mapped on walking maps – available in shops on the island).
Unless you have good local knowledge, I’d avoid most of the ‘goat tracks’ as they are often dead ends and you end up in an olive grove and have to retrace your steps!
I was there last month for a family wedding and the island is actively encouraging walkers – quite a few clear way-marked walking routes have been introduced, however few of them would be suitable for a mountain bike and access rights seem a bit unclear …
We did meet a large group of German tourists on mtbs, hybrids & road bikes heading out on the road from Argostoli to Mytos beach. The road north is closed Above Myrtos though due to an earthquake causing a land-slip earlier this year. On the whole though there isn’t much provision for cycling so you’d have to be pretty self sufficient with spares etc.
A XC bike would be ideal and you could mix and match the ‘white roads’ and the dirt roads. It gets very hot though so early morning rides (like you’ve suggested) are often best.
It’s actually pretty mountainous so there’s a lot of climbing involved – even just to get from one village to another. I went out early in the mornings and stopped off at the bakers on the way back to the villa with fresh bread for breakfast!
Nearby Levkas had quite an MTB scene a few years ago – and was favoured by the likes of Neilson Active holidays. This was before the boom in the Alps and MTB was more of a ‘value-add’ to sailing & windsurfing holidays.
Hope this helps … Enjoy!