OP there has been a lot of suggestions here for things that you can do to improve the nature of your relationship with your DD, without the need to go to court. I'm another who doesn't 'buy' the notion that your life is so busy that you cannot increase the amount of time/frequency you see your DD. You need to seriously look at how you organise your life to enable you to increase the contact you have at the moment. And if you really want to play a more significant role in your DD's life, you will need to change things as they stand at the moment.
As has been said, you can internet shop - deliveries can still come to your home very late at night so perfectly possible to arrange that for after your contact with your DD. So there is one extra slot per week you could manage with your DD. Washing/ironing? Really? You can't start that after you see your DD for 2 hours another evening after work? My DD goes to bed at 8pm, sometimes half past or 9 if it's a late night. I still have to sort washing after she's in bed, and usually do ironing in the morning before we get ready. Or, I'll iron the lot in one go at the weekend, after she's in bed so again, it starts at about half 8/9pm. Doctors appointments - how often does that happen to interfere with your weekly life to be something that you need to factor into a contact schedule with your DD?
I do actually feel for you, as I know it must be frustrating to not be able to do more with/for your DD, but the crux here is, it's in your hands to do something about that if you are genuinely committed to improving the relationship with your DD, as opposed to taking some of the perceived control your ex has in this situation. The reality here is, your ex is your DD's sole carer, and has total responsibility for everything your DD needs and wants. She has that role in your DD's life because you have been complicit in allowing that to develop by virtue of your apparent 'busy life' and going awol for a month. You need to realise that you had choices here, and if your aim was to play a more significant role in your DD's life, you made the wrong choices. It's not a lost cause though, and you have choices now that can help to turn things around. Your 1st priority has to be that you increase how often you see your DD every week, and you have it in your hands to do that. And yes, that does mean putting spending extra time with your DD above spending that with friends/family. If you make the choice now, that you are still too busy to fit more time into your week to see your DD, and initiate the court process, I have to say I'm not optomistic that your relationship with your DD will improve in the way you want, or to the timescale you want either.
You could approach this by asking your ex to talk, explain that you would like to progress to time alone but understand the issues with separation anxiety. In order to address that, you think it would be better if you spent more frequent time with your DD, and propose extra nights/days that you can fit in (after going through your week as above, organising all these 'busy' tasks better so you have some more free time) and ask her to agree to a period of time where this can help build up towards you having time alone. Maybe suggest she moves away, out of your DD's view but remain in the same room, so she can see how at ease your DD is with you, as you build up that contact. Your role here is to keep your DD occupied, keep her focus on you and what you are doing (playing a game she likes, with her favourite toys etc) and demonstrate that you can keep her happy and content even if her mum is not within her view. The key to keeping a child who has separation anxiety calm (in my experience, albeit my DD wasn't too bad) is to distract them. You need to learn ways to do that, so it lets your ex see that you can cope, and when your DD shows signs of getting upset, you recognise them, and are able to react in a way that heads off the tears, or manage the tears so that the don't last too long. It's not easy! And you also need to understand that when your DD cries with that separation anxiety, that does have a physical effect on your ex too, which is probably why she cannot bear to let her DD cry for long if she starts when you are with her. It is hard to describe but I can still remember how my DD's distress made me incredibly anxious so please bear that in mind when you are getting frustrated with your ex and her unwillingness to let you try and comfort your DD.
I hope you take on board the many suggestions and really think about what you can do to improve things, without the need or risk of going to court to change things. Ultimately your choice, but I hope you can improve things without the need for court.