I tend to be one of those people who moves at a faster pace than some, so I get it. It's just very few attractions are that close together and you need to give yourself time to get from one location to another. Even if you take taxis everywhere (which will really add up, things still take some time.)
If it were me... I would probably:
Day 1 May 4 Take a 7:03 Shinkansen from Tokyo to Hiroshima. You will need to transfer in Shin-Kobe, but it's an easy transfer and taking an early shinkansen like that might give you a nice view of Fuji-san. I know others have not had this experience, but I've gotten some stunning views in May when I am up early enough.
You'll get to Hiroshima ~11:35 and can go find some lunch. See what you want to see in Hiroshima over night in Hiroshima. With your budget I would stay someplace like the Sansui Ryokan or Chisun Hotel. Both should be within your budget. I am pretty sure I've stayed at a Chisun at one point. They're a pretty basic business hotel.
Day 2 May 5 go to Miyajima, come back and do anything you want to do in Hiroshima, night in Hiroshima or take a late (19:23 post dinner) shinkansen back to Osaka. There are lots of options in your price range, just make sure to pick somewhere you can check in at 22:00. Since you are two adults I would probable go onto Osaka, just because taking a 19:23 or 19:24 shinkansen puts you in a better location for Day 3, where you're not wasting daylight hours. Also, if you finish earlier than expected in Hiroshima, you're where you want to be.
Day 3 May 6 If you're still in Hiroshima, take a morning shinkansen back to Osaka and see the aquarium and then take an afternoon train to Nara (night Nara) If you're in Osaka, go to the aquarium and then go to Nara (night in Nara) If you do the night in Osaka on Day 2, then Dotembori then, otherwise skip it or visit when you are in Kyoto, it's really not that interesting. Outside the Nara hotel and the Kasuga Hotel, there should be quite a bit in your price range.
Day 4 May 7 see whatever you still want to see in Nara and then take the train up to Kyoto and see what you have time for in Kyoto (night Kyoto)
Day 5 May 8 see what you want in Kyoto (night Kyoto)
Day 6 May 9 Leave from Kyoto for Kawaguchiko.
This itinerary isn't so different from your original one, but by starting in Hiroshima, I think you get a little more time in each destination and get around the fact the Friday and Saturday of Golden Week are relatively booked and many hotels have increased their prices for it.
You should find a lot in Kyoto in your price range with this change.
I've stayed in the Dormy Premium for 1 night. The room was small, but the staff was lovely, it was near Kyoto Station and the baths were pleasant it's $265 for 2 nights
There are a large number of places that are under $200 for 2 nights, like two of the Mitsuis which seem to be in nice locations or spend $225 for the Citadines Karasuma Gojo, which is generally well liked. I would not take anything larger than a purse on a Kyoto City bus, so I would pick someplace to stay you can walk to and/or take a subway or train to. Being near a bus stop is great for sightseeing, but not when you have bags.
Many people would consider this a fast paced itinerary, but it should give you a chance to do everything you wanted to do outside of USJ/Koya-san and has the flexibility if you're moving faster to add some more. You won't be able to add Koya-san, it's a 2 hour+ one way from Osaka, but I guess if you wanted to splurge you could maybe squeeze in an hour or three in USJ. I would not do it though unless you have Express Passes or are arriving before park opening. I went on a "quiet" day and by 11:00 am most rides had a 40 minute or longer line outside rides explicitly meant just for children and you would be going during Golden Week. There was a 1 hour wait for Jaws by 13:00. I don't want to know how long the line for Harry Potter was.
If you don't mind paying out of pocket, can sleep on a bus, and didn't have big plans for the day you arrive in Kawaguchiko, you can always take the night bus between Kyoto Station and Kawaguchiko.
I'm too old for night buses, but many budgetary travelers love them. It would give you a little longer in Kyoto.http://bus-en.fujikyu.co.jp/highway/detail/id/9/