Centrelink payments are also asset tested, so they will take things into account like money in the bank and cars, etc. And if you've earned well this financial year you might not qualify for much in terms of FTB A/B, but if you stop work early in the next financial year (in a few weeks) it's different.
Have you thought about getting a morning nanny perhaps? Maybe just for a few days a week, but the idea is that you hire someone to come from say 6:30am (just before you go to work) til say 8:30am (whatever is reasonable for your daughter to wake up & have breakfast) and then take her to childcare. You might even find an older lady who is an early riser and would be happy to do that. They do exist, I know of at least one person who uses it, for exactly the same reasons you state (needing to get to work & 6am is an early start for little ones).
Long days apart and long days at childcare suck and it's hard work all around. I did it for 3 years and it was extremely hard, especially when they are little (and I wasn't single so I could share the load). Now, I work part time (2 - 3 days a week and not full days) and it's much easier to manage. It still takes a fair bit of organisation and pre-planning and the general morning chaos, but I'm thankful for those mornings when all I need to do is take my ds to school & make sure my dd (20 months old) is dressed for the school run.
I'd suggest you take your info to centrelink and speak to someone about what YOU would actually get if you were to stay home full time (next financial year to keep it simpler). Or you look at part time work options, possibly just including shorter days (finishing 1 hour earlier could make a big difference), or 4 weekdays instead of 5. Or look into other care options for the start & end of the day to take the load off you a bit.
Hope you can sort out something different soon. It is hard.