I'd echo some of the above. It is going to be pretty hot and uncomfortable in July. If possible, try to get the early train from Hiroshima to Himeji, which only takes about an hour, and see the castle in the morning. Then spend the afternoon on the air-conditioned shinkansen going back to Tokyo.

Miyajima is on an island in the bay outside of Hiroshima. You COULD use a cab to get to and from the Hiroshima Station to the ferry pier but it is far less expensive and simpler to just use the local train. Your rail passes will cover the rides and you don't even need tickets. Just show the Pass. If you are concerned about luggage on the local trains, just have your previous hotel ship your bags to your next Tokyo hotel by takkyubin service and carry enough clothes and stuff for the Miyajima stay in an overnight bag.

If you opt to leave Miyajima/Hiroshima later in the day, there are a few decent hotels in Himeji within short walking distance of the castle. A later check-in, evening stroll through the artsy shopping street next to the main drag and an early-morning visit to the castle would still let you use the next afternoon to finish the run back to Tokyo. For "nice" accommodations in Himeji, there is the Hotel Nikko adjacent to the station (around $210.00+ for a double) and for the budget-minded, the American "Comfort Hotel" chain has a new-ish "nice" business hotel near the station that is about half that. I frequently use Comforts in Japan and find them comfortable, clean and useful, if a bit sterile. They include decently sized rooms, free WiFi and light breakfast but rarely have an on-site restaurant for other meals.

There are a few departures from Himeji Station that go direct to Tokyo but most will require a change at Shin-Osaka.