I do understand where you're coming from.
There was a point in my relationship at around 5-6 months where, because of the amount of time my BF spends with his DD, I thought to myself that if he didn't start making moves to integrate me more into what I called 'his other life', then I would probably have to start evaluating whether this was a situation I could live with long term.
Luckily it didn't come to that because shortly after he suggested introducing me to his DD which I was obviously really excited about.
However, it was still a good 6 weeks between him telling me he was going to introduce me and me actually meeting her. The timing and circumstances needed to be right.
What I'm saying is that, even him just saying he wanted me to meet his DD was enough of a statement of intent for me. I was then happy to wait another month or so before it actually happened.
I'm still not really understanding why you have this sense of urgency. If you have talked with your GF about her meeting your DD then why does it have to happen immediately? What do you mean by 'life passing you by'? Is your GF putting pressure on to speed up the process?
I'm only asking because I get the sense from your posts that you're focussing more on what would work best for you and your GF and that your DD will just go along with whatever because hopefully she won't notice.
Obviously ultimately you can do what you want but I agree that limiting visits to a few hours during the day and holding back on affection in front of her would be the right way to go for now.
Also, I will say this: you are used to having a relationship and a child and are prepared for how having a child impacts on your relationship. Your GF won't be. I am not and am finding it a steep learning curve being with BF and his DD and us not being able to be as spontaneous as we want to be, take off for a weekend, or stay in bed all day if we want.
Those things are very important in a new relationship so don't be in a hurry to bypass that wonderful courtship period in order to skip straight to happy families.