I graduated a while back with a Bachelor of Science (Advanced Mathematics) degree from USYD. The process of transferring shouldn't be too difficult. If you get reasonable marks in first year, you'll be able to transfer.
For USYD, the only difference between the Bachelor of Science (Advanced Mathematics) and Bachelor of Science (majoring in Mathematics) is just the name. Aside from that, there is 0 difference between them. In my opinion, the "Advanced Mathematics" degree is just a marketing ploy to attract students - the cut-off for the Advanced Mathematics degree at USYD was 99.1 (IIRC) in my year, so it attracted all the top math kids (I was in the same classes with the entire Aus IMO team).
I've attached an image from my degree, and as you can see, it just has the words "Advanced Mathematics" tacked on at the end. Aside from that, there's no differences between the two degrees when you graduate.
When it comes to getting a job, it doesn't make a difference whether you were in the "Advanced Mathematics" degree or not. The only thing that matters is that you have a major in math. So my advice is not to fret if you don't get in - just do a math major in the normal Sci degree and it literally makes no difference.