I would save the entire game state with Kryo between turns. This is very likely fast enough for a turn-based game on mobile. On desktop it is fast enough until you are saving 7-8MB of data. Kryo is binary and the data is small, so this is quite a bit of data. Saving as JSON with libgdx might be ok, but being a text format it has never been intended for high performance. You could look at jackson for JSON if it is too slow and you want human readable.
Kyro does object serialization, correct? Wouldn't this mean rhat the format of the save file would be directly dependent on the layout of my objects? So, if I update the layout of my objects then existing save files would become invalid? This would be true of any object serialization scheme. I remember reading about such a concern somewhere, might've been here.
Still, if Kyro is fast enough that I can just brute force serialize everything on every turn (actually I'd still make sure I only serialize things that have changed) it might be worth the hassle of managing outdated save files (which may not even be a problem for my game anyway).