Hi, the high end soaplands normally charge a 入浴料 (charge for the bath/nyuyoku ryo), and the サービス料 (girl's charge/service ryo) is separate.
The medium to budget end soaplands will charge you a 総額 (total cost/sogaku).
Syncrogue’s suggestion about saying you're 日系人 (nikkei jin) is a good one.
Here is an excellent guide to soaplands in Tokyo. Have fun with your next trip!
https://tokyoadultguide.com/thre ... /page-12#post-80996