Feel for you mate - been there. The good news is you give a shit and you're trying to get it right - which means you very probably will.
Heartily agree with the above advice. I'm surprised the court order didn't give you "overnights" at least every other weekend (which you can then be flexible about with the Mother). I would start by explaining your concerns to the Mother and asking her to agree to overnights on Friday or Saturday and if she says no just leave it and bring it up again in a few weeks. If she's really stubborn after a couple of months I'd go back to court. Notabedofroses is right when he talks about building routines, making dinner etc - it shows that you are an everyday Dad who does everyday Dad stuff and not just fun Dad who's great for trips to the park but not the serious stuff - and the serious stuff kicks in pretty quickly. He's also right about "door firmly open" and "stages of childhood". There will (there definitely will) come a day when, after a row or a refusal, she comes running to you claiming she has the worse Mother in the world and can she stay with you, live with you, never go home... There will be times when she wants to talk to you and not Mum about things in her life. There will be times when you are the go-to-guy. Trust me.
At age three kids can be pretty random and I don't think you've got anything to worry about. You've got probably fifty or sixty years of being Dad ahead of you ! Also, kids are hard work and I would be surprised if Mum wasn't keen on some time off - it's a just a question of trust and confidence and if you've never been in a relationship with the Mother then that can be hard to assume / establish / build. Time is everything. People chill out and calm down over time and it will all click into place. You have to be constant, reliable, honest, available, ready to be flexible and it can be (is) frustrating when progress seems slow but just keep on keeping on. It is though vitally important that you establish yourself in your daughters eyes as DAD in every real wy, especially in the Mother starts a new relationship which you have to be ready for. I would say overnight stays are absolutely key - maybe not this week but sooner rather than later.
Keep the Mother sweet, do the constant love, support and guidance bit and look forward to the relationship with your daughter deepening as she gets older. Always tell her she has two homes, two places of safety and as she starts nursery and then school make sure you are also listed as a contact and get all the notices and letters home (this is really important) and that you are also listed as an emergency contact.
I split with my ex when our son was five months old and it was all very nasty. At four years he was staying for weekends and by the time he started school the ex and I were doing parents evening together and he was coming away with me on foreign holidays.
You sound like a good bloke and the Mother will recognise it. Suggest some overnights (don't push too hard) but keep at it. It will work for everyday Dad stuff and it will help your daughter understand that your home is slam her home.
Good luck : )