For your first invoice, transparently detail the hours you spent on different tasks. Be honest about the time invested in development and highlight the value you've provided.
When explaining the product design costs, emphasize its crucial role in enhancing user experience and overall project success. Clearly outline the benefits and impact of the designer's work.
Approach the rate renegotiation with the product designer openly. Share your budget constraints, express the value of their contribution, and discuss a rate adjustment that aligns with the project's financial dynamics.
I wouldn’t “mask” the product design costs, clear, honest communication is important.