Wow- I am surprised. They must be convinced you aren't Japanese (or maybe underage) if they are asking you for ID. That is kind of rude, and I wouldn't expect that kind of treatment from the highest end places.
That being said, it is easy enough to make a fake health insurance card. Since it has no photo, and no signature, it's perfect for your mongering purposes. You can literally scan your real card, photoshop a kanji name into it, photoshop out the anti-copy stuff, print it, laminate it, and there you go. In the dark, its probably good enough.
I'd recommend NOT going to a soapland in the same prefecture as your health insurance card, as the tout would probably be able to tell its fake. But if you show a Kyoto tout a Tokyo health insurance card, he won't suspect a thing, because he probably has no idea what the real Tokyo health insurance card looks like.
And if they still make a stink, screw it and go to somewhere else where they value your patronage.