I agree with a previous poster that this isn't sustainable. I think you need to look at reducing your outgoings so that you can have regular weekend days off.
E.G. You are paying 1200 in weekend childcare out of total income/outgoings of 5500 a month. Is weekend work very much more lucrative than weekday work? If not, then working 5 days a week instead of 7 would lose you income of 1500, but reduce your costs by 1200. ie you are only 300 better off a month for working all your weekends! Two weekends off a month would only cost you 150. Can you reduce your other outgoings to cover that?
Ask the daycare to pass a letter to the other parents to see if anyone would like to liftshare, with you contributing to the travel cost. 700 a month on taxis is a lot!
Maybe look at moving to reduce rent and/or travel costs to daycare.
Maybe there's another parent you work with who you could do childcare swaps with - eg you each take both kids for one weekend while the other works. Ie each month you have one weekend working (which earns you 300) one weekend looking after both children, and 2 weekends off with your daughter. This actually leaves you with the same income as at present.
If you are worrying that your daughter will miss out on camping, amusement parks etc if she isn't with the weekend sitters, then don't!! She'd much rather have some time with you, especially at this age.
(As an aside: personally I'd prioritise having weekends off for a few months over a single 2 week break, but that's obviously personal choice. But do be careful not to build so much expectation on those 2 weeks that it overwhelms you both, and you end up getting upset at each other)