Certainly, the advice you have been given about prepaid phones is technically accurate. Without a 外国人登録証明書 (alien registration card), officially, it is not possible to purchase a prepaid (or postpaid) phone in Japan.

However, last April, I walked into a random cell phone shop in Higashi-Shinjuku and signed up for a standard, post-paid SoftBank "White Plan" with 1-year contract using only my passport and credit card, no questions asked. I'd imagine that the experience purchasing a prepaid phone would have been similar.

You're likely to be better off at smaller shops that aren't necessarily trained in the "proper" way to deal with foreigners, but this will necessitate at least some Japanese ability since most such places are unlikely to have English-speaking staff. In any case, purchasing a phone without an ARC is definitely doable, even if not officially permitted.

Finally, depending on how long you will be in Japan, there are reports of people who have been able to get an ARC without a long-term visa if you want to do things the official way. YMMV, of course.