One of my favourite birding experiences (though i went a bit later, in early April).
The only "cheap" option IMHO are the youth hostels - they approached UK prices, i think i paid £15 dorm bed only in the one near to Karuizawa (Honshu), i was the only person there - that was about 7 years ago though. There might even be independent backpacker hostels, which are usually cheaper still, though vary in quality from v v good to terrible in each country. At least there's Trip Advisor to check first these days.
Minshuku were £30 - £50 pppn. The one at Washi no Yado is a fantastic experience for the Blakistons & not too pricey, the chap at Furen Lake B&B can book it for you. He also has a friend who has a B&B nearby which is half the price [of his]...
Went from the car hire at Kushiro Airport to the ferry at Tomakomai via the last train, which saved on an overnight.
I bought a Shinkansen pass, only available in the UK, which was v cheap compared to Japanese prices, you can get a plane pass also which may be practical with a family?
There are the odd "local" overnight trains where you don't have to buy a berth - i caught one on Kyushu to the Japanese Murrelet site but it you can't really lie down anywhere. Incredibly slow too which is agonising if u wanna get there and find a park bench!
B interested to hear your experiences (eg overnight buses), as i'm really keen to get back there.
Strongly suggest you join "Kantori", the Ennglish-speaking Yahoo group about Japanese birding, it's very helpful, informative and polite... but what would you expect lol?
Sure you'll have a fantastic trip.
Edit - flights on net advertised Manchester - Sapporo for £393. Somehow i doubt it though.....