Headed back to Osaka tomorrow. The plan is to stay 2 nights so we can visit Kyoto and then head to Hiroshima.
Would I be better off staying in Kyoto or just finding a convenient place near Shin-Osaka instead?
As other said: with a JR Pass the distance between Shin-Osaka and Kyoto (station) is moot.
On the other hand, the good parts of both Kyoto and Osaka are quite far from their stations.
Twice I had to take the same decision, and twice I found that rooms near Shin-Osaka are considerably cheaper than, say, in Gion, with the additional goodness that Shin-Osaka is convenient to move around with the shinkansen (for you, notably to get to Hiroshima).
Last time, I stayed here: https://www.airbnb.com/rooms/4486915
The apartment is small and very clean, but it's quite near to Shin-Osaka station and you don't need to cross any overpass, so moving around with heavy bags is easier. Also the closest entrance is the central one, where you get in for the shinkansen, so it's twice the convenience.
Oh, and the room has a washing machine inside, so you could even wash your stuff there (but probably 2 days aren't enough anymore for your clothes to dry, though).