That is very useful information, tokyo_girl!
As for refusing non-Japanese, it's true to say that some exclusive (very expensive) ryokans and restaurants just refuse anyone not known to them or not introduced by existing clients. Japanese expression is Ichigen san okotowari 一見さんお断り。This is regardless of nationality, but since most foreigners are visitors and are first-time clients, refusal can be common. I'd say this is becoming less, with the huge increase in overseas tourists and the money they bring to Japanese economy. Many businesses rely on them to survive. As for refusal for not speaking Japanese, again things are changing and foreign faces are becoming more common even in remote areas, and those in hospitality industry are increasingly employing staff with foreign language skills (English, Chinese, Korean etc) in a highly competitive market.
I have used the Japanese Rakuten site (and other booking engines like JTB, Nippon Travel, Jalan etc) and never had any trouble. I do find that room offers are very similar across sites and often the best deals are on the hotel's own Japanese website, and some hotels guarantee to beat any other offers.

Alternatively, if anyone has any good tips for finding cheap accommodation for 2 people in the country, I'd be happy to hear it.
Love Hotels can actually be a good deal. Cheap ones cost around the same you'd pay for a hostel dorm (but here you get a rather large room with attached bathroom). On the minus side, they don't offer multi-day stays, so you have to deal with luggage somehow. On the plus side, they often have space even in peak holiday periods.
I tried it, it worked perfectly. I didn't need a Japanese credit card, or phone number, or anything really, apart from the ability to understand the website. A useful find.
Next up I suppose, an adventure in a love hotel at some point...
And you're absolutely spot on tokyo_girl, local government tourist websites do often have useful information on accommodation (including campgrounds!), and I often use them as well.
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