So what are you going to do about it now?
If you (your DP) think it's not ideal but DSD will survive.
- Try to get DM to change her ways.
That's not very likely but perhaps worth a try? Have a think on how you might increase your chances of achieving this. Obviously the way you have been approaching it so far didn't work. Perhaps it has got to be only your DP who talks to her, rather than you. Perhaps you need to be super careful not to raise her defensiveness. Focus on the one thing you most wish to change rather than have a whole list of demands. Always voice it as concern for DSD rather than which grown-ups have what rights. Is there someone she'd be more likely to listen to? Perhaps a grandparent?
- Suggest to change the arrangement in a manner so that DSD is with you more frequently on weekends, with DM on weekdays.
You aren't keen on this but if you think it would help, then perhaps put the child first and just remember, it won't be forever. I know you wrote that it wouldn't make any difference if DM had her weekends/week days, so it would not actually improve the situation for DSD whilst still making things more difficult for you. But later you implied that it IS worse for DSD on weekends with DM, compared to weekdays. DM still goes out on week days but less frequently. And perhaps that boyfriend stays over less frequently on week days?
I know it is unfair; DM can basically blackmail you into accepting whatever arrangement suits her best. If you don't comply, it is DSD who suffers. So your choice is if you want to stay principled and refuse to be blackmailed and accept DSD's welfare as collateral damage, or if you choose to put DSD's needs first.
- Try to have DSD with you more.
I understand you would prefer this, as then DSD would be safe and cared for, and you'd still have the freedom of occasionally using other childcare and have a bit of social life. Also you say DM doesn't want this as any such arrangement would involve less money for her.
Perhaps your chances of achieving this are best if you do it gradually. Maybe aim, as a final goal, towards having DSD with you 5 nights/week or 11 nights/2 weeks, to include all weekends? Keeping some flexibility so that you can swap days e.g. if DM has some special plans for doing something with DSD on a weekend. But to get there, start by suggesting just one more night/fortnight with you (preferably a weekend night). Show DM that it doesn't make that much difference financially, just one more night out of 14. Then after a couple of months or so suggest adding another night. And later another - then you've reached 10 nights with you, 4 with her.
Again if you attempt to do this, think hard about how you can maximise your chances of achieving your aims. Who has to do the negotiating? Is there a go-between, someone who'd be able to suggest the idea to her without her becoming defensive? Perhaps you could find a Saturday activity for DSD that DSD would really love, but DM can't take her to, so the only reason you propose having DSD with you every Friday night (rather than every other Friday night, as it is currently), is so you can take DSD to this activity that she loves? Most of all you need to avoid the message 'we want DSD more with us because we think you don't look after her properly' which is 100% certain to raise her defensiveness. And you need to make sure it's not about the money either, that she doesn't think 'they just want her for an extra night so they can cut my maintenance payments'.
So I think you and your DP need to sit down together, think really hard about how bad it actually is, how much you want to change it and what you are prepared to give up for that. Then decide on what you'd like to achieve. Then think about how you need to go about things in order to achieve that.