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.

Just as there are advantages to starting your own business, there are many many disadvantages. Jobs provide reliability and stability, and almost always are less work than starting your own business, and can provide a high income.

Put it into a high yield savings account, they are yielding 4-5% right now, at least until you come up with something else you’d like to use it for.

There’s certainly many jobs where remote work isn’t possible (at least part of the time). E.g. Those who need to be in a lab/shop. Even a seemingly non-technical job, like a salesperson, may need to be in person for meetings with clients, etc.