Is your immediate desire:

A) A job where your title is 'Senior Product Engineer'?

B) A job where your responsibilities and activities are that of a 'Senior Product Engineer'?

C) A job where you can hone your craft over time, so that one day an impartial observer would consider you a 'Senior Product Engineer'?

D) Something else?

From your description, my guess is:

A is achievable. Many companies have a low bar and/or are willing to take chances and/or are happy to grant big titles because they don't cost anything.

B is achievable, but you haven't built the skills or experience for it. So it's probably not the ideal learning environment and would probably cause a lot of stress.

C is what I would recommend if a friend came to me with this question.

(If you're a good PM, this will save time on some aspects of your new engineering job, so you'll be in a better position than a typical junior engineer.)