If she would leave you for something as harmless as demanding to see the phone, then she doesn't sound like the kind of person worth pursuing. Obviously, if she cares more about you looking at a damn phone than your relationshop, it's not much of a relatiponship anyway.
Also--and this next part I am very passionate about--you better make up your mind right now what you're going to do, as you are fucking with the heart & mind of an 11 month old child who has known you since she was 3 months old. The longer this relationship goes on, the more potential harm the child will suffer when the relationship ends. It would be much more injurious if you waited until she was like 2 years old than ending it now. That would cause severe harm to the child...and guess what, it would be YOUR fault because you didn't deal with shit when you had the opportunity.
Whenever there are kids involved, it's no longer about yourself. You need to put the child first regardless of whether they belong to you or not. Sorry to say it, but the child's well being is more important than yours...and neither did the child ask for you to come into her life, so you better figure out your shit quick.
Personally, I think it is IDIOTIC for any mother/father to bring another parental figure into the home until you are either 1) Married or 2) Been together a long time and have stable relaionship with the intention of marriage
The fact this woman let you move into her home when her baby was only 3 months old is absolutely fucked and shows you right up front she is either not capable of prioritizing the wellbeing of her child or two, that she willingly places her own desires first. Good parents don't allow any type of strong bond to form between their significant other and their child unless both people are confident the relationship is moving towards marriage and even then, interaction between the child and significant other usually progresses very slowly, thereby minimizing potential risk to the child in the event of a break-up.
Not only is it extremely irresponsible and careless to thrust yourself into a child's life during a new relationship, but to do so when you already know there are glaring problems in the relationship. Well, that's is a clear sign that neither adult has the child's best interests at heart.