If you want to stay overnight on Miyajima Island, you might want to book ahead. There are limited spaces and it's often full.
As for seeing Ise and the beach in Mie prefecture and then going to Koya-san, it should be doable, but you shouldn't need to head all the way back to Osaka. According to some maps I found online, there are buses from Mie prefecture to Gojo in Nara prefecture, which is 1 stop away from Hashimoto in Wakayama prefecture, which is at the base of Koya-san. Just change trains in Hashimoto and from there you'll be in Koya-san in about 35-40 minutes. You can also take a train to Nara, and from there go through the Yoshino valley to Hashimoto.
http://www.tb-kumano.jp/en/transport...tion-Times.pdf
http://www.tb-kumano.jp/en/transport...cess-chart.pdf
Finally, if you want to take a while longer to explore the Kii peninsula, from Mie prefecture you could take the scenic train ride around the whole thing. There are lots of places to stop if you are interested, from the tallest continually flowing waterfall in Japan at Nachi Katsura, to the whaling museum at Taiji, to the white sand beach at Shirahama, to the rocks marching into the ocean at Kushimoto, etc. The area is famous for its produce and in the right seasons you can get fresh mikan oranges, homemade soy sauce, Japanese ume plums, ume wine, and salted umeboshi, etc. Ride all the way up the line to Wakayama City, and then head east to Hashimoto from there.
Oh, definitely stay at a temple while you're in Koya-san, there are many to choose from.