Thanks again everyone for your input!
I have been vacillating over where to go for my day 1 and 3, and day 2 will just be a Hiroshima + Miyajima day trip. On day 1 we will arrive at KIX around 10AM, and I was planning on going straight to Kyoto station, eat lunch and store our luggages there, and see the bamboo groves and fushimi inari (both accessible by JR trains) before checking in at an AirBnB in Shin-Osaka. Afterwards we could maybe go to Dotonbori or Kyoto again for the evening.
An alternative would be to go to Himeji for half the day and eat at Wakkoqu in Kobe before checking in at the AirBnB in Shin-Osaka, then proceed to Dotonbori or Kyoto for the evening if time permits. However, I do not know how crowded the castle would be and/or if keep tickets would be available. I have read that the keep is boring though, so maybe just walking around the perimeter of the keep would suffice?
So this is the overview for the Kansai leg of the trip:
Day 1 (Mon): Arrive KIX, Himeji or Kyoto (bamboo groves + Fushimi inari), stay in Shin-Osaka
Day 2 (Tue): Miyajima + Hiroshima, stay in Shin-Osaka (discussed earlier)
Day 3 (Wed): Himeji half day + Kyoto half day or Kyoto full day (Higashiyama area), stay in Kyoto
Day 4 (Thu): Kyoto half day (Kinkakuji, just roaming around the city) then Tokyo
Thanks again!