Usual disclaimer: I am not a lawyer and this is not legal advice.
What the institute says is completely wrong: there is no such law.
The salary of Wissenschaftliche Hilfskräfte (HiWi, student doing scientific work) varies by state and by university, and it is not regulated by law. As explained in Chapters 3.3 and 6 of this document (unfortunately, information about this is only available in German), these student employees are in most cases not even included in the worker unions, and tariff negotiations do not apply to them (the only exception being Berlin). This means that these salaries are almost exclusively decided by the employers.
The only available "official document" are the Richtlinien der TdL über die Arbeitsbedingungen der Hilfskräfte (Guidelines of the Tariff Association of German States about the working conditions of the assistants), which are just guidelines, not a law. These guidelines only provide a maximum salary (not a minimum), and distinguish between three groups of students. The highest salary group is for students with a Master's degree. The second-highest salary group is for students who, among other options, have a
i.e., a Bachelor's degree. There is no clause that says that currently studying another Bachelor's degree disqualifies you from being in this group.
This said, getting your foreign Bachelor's degree recognized might take some effort, but I would at least ask them what law they are talking about.