Maybe your original description is incomplete or I'm not understanding it correctly, but I don't see a problem here, other than that you and they didn't have a proper mutual understanding up front of your and their business relationship, and particularly regulatory responsibility.
At your end, that shortcoming would call for better contract review, and a
Sales department that pays attention to regulatory responsibility as an essential parameter of every sale. Your customer needs to improve their purchasing in that regard as well, of course.
I assume from your original description that you thought you would be the regulatorily responsible Manufacturer, and your customer would be a distributor without regulatory responsibility. It sounds as if your customer thought...either up front, or as an in-course change...that they would be the regulatorily responsible Manufacturer, and you were performing secondarily-responsible contract manufacturing for them. The labeling approach they now want is consistent with that second scenario, which is a regulatorily acceptable approach, and in fact quite common.
The fast dancing you need to do to transition from the prior plan to the new plan is no big deal. Just document what you do and why it was necessary.
In a contract manufacturing role, you have fewer responsibilities, which you might view as meaning lesser costs. If you make the same revenue, it's not a bad way to do business.