@User#8628 has explained it already, but I am not sure if you understand the fee structure of a soap - bathing fee for the shop and service fee for the girl, usually in the ratio of 1 to 2. With all my respect to your experience and expertise, I would like to point out several points.
A shop has to deal with two markets. One with the girls and the other with the clients. There are interconnected two markets the shop tries to match to maximize its profits. As Alice explained, some girls are reluctant to play with non-Japanese clients. I know this is a result of blatant racial prejudice because even those girls are OK to take any Japanese even if he is a full of shit. Even in this setting, some shops want to make more girls available to a wider client base including non-Japanese, overcoming the difficulties. The gaijin tax works as an incentive for the providers' side, which makes more girls available for a wider client base, and ultimately improves the profit.
Also, I have to point out that a discount is given basically for "free clients (those who don't nominate in advance)" who are available for girls not busy or even unpopular. Having a girl redundant, just waiting for clients in the waiting room in a shop, is costly for both the shop and the girl. Many local mongers do not walk up to Yoshiwara and negotiate. They usually book in advance to get complimentary shuttle services from train stations nearby, say, Nippori or Minowa or Ueno, even if they don't nominate any. Some of them are motivated to find girls they want to repeat. Truly popular girls are very difficult to book for first-timers.
As I understand, the best strategy for a shop is to retain popular girls and repeating clients. There is no incentive for a shop to give a discount in the way you wish to be.