At least on the way in I wouldn't want to stay in Itami Airport.

You can either
a) Get a direct shuttle bus to Kyoto station that night
or
b) Get a 20 - 30 minute shuttle bus to Umeda and stay in one of the hotels in Umeda one of the bus stops plops you directly outside the Hilton Osaka (a much nicer place to stay than an airport hotel....) (you'll have plenty of choice of food in Umedia) In the morning you can then take the Shinkaizoku train from Osaka station to Kyoto in the morning which runs regularly and is fast)

As for the way back checking in at 6am is early - Japanese domestic airports are VERY efficient and if you have showed up on time they will get on the plane - that notwithstanding that is the main rush hour but even so you could stay at a hotel in Umedia and get a bus to Itami at 6am and still arrive in plenty of time for your flight? For example the first bus from the Hilton leaves at 05:50 and would get you to the airport by 06:20 ? https://www.okkbus.co.jp/timetable/U/#go