A father automatically gains PR when he is married to the mother. The other way to get PR is to be registered on the birth certificate after December 2003. Before this time, then you won''t have PR. You can obtain this with her consent and go into a PR Agreement with her. If she doesn''t agree to this, then you can apply to the court for PR Order. The courts will likely give this to you as they do encourage this for fathers. Once you have PR you will then be entitled to apply for S8 orders, such as
contact / shared
residence if you needed to in the future.
From what I have read, you sound like an excellent father, and the chances are very high that you would get a PR order in your favour. If you can get PR then this will give you equal rights and responsibilities to the mother in the eyes of the law.