Since you say that transferring (which sadly would be by far the best option, see the note at the bottom) is not feasible, the next best solution might be to just will those courses into existence. From your question I get that you haven't done the available courses yet, so start doing those as soon as possible. Most of them are soft prerequisites for the more advanced courses anyway. Try to excel in those courses, in order to get noticed as a bright and interested student. (I assume that you are good enough to do so, otherwise going to grad-school will be a bad idea.)
Then, once you are finished with some of those courses, talk to the professor and tell them what you told us. Make sure that the focus is on you being interested in learning about things, not in padding your transcript (which is just the means to study more math in graduate school). Teaching the same old basic level math year in and year out to students only interested in their grades can be quite boring, so at least some should jump at the chance to break the monotony.
From there on, I see several different options you might discuss, in rough order of desireabilty:
- The course you want is actually created, at least as a one-off. This is more likely if you find a few other people which are interested as well, but don't be disappointed if it doesn't happen, as preparing and holding a new course takes quite a bit of time which simply no professor might have.
- You do the advanced topics as a "reading course". This is not a common format, but the basic idea is that the professor assigns you a textbook to read and you periodically meet up to discuss your understanding and any problems you might have had. Then in the end you do an oral exam to determine a grade. I'm not sure about Australia but at least here in Germany this can count as equivalent to a regular course and is sometimes done in this exact situation if there aren't many interested students and the prof has not enough time.
- The prof might just assist you in studying the topics on you own, by giving you some hints and opportunities to periodically discuss things together. Sadly you will not have formally passed a course, but this might be the only solution in the not completely unlikely case that bureaucracy simply doesn't allow for a certain course to exists in your university.
In any case, an important goal will be to get a good personal letter of recommendation. This won't get you to your top choice of place but could at least open up some possibilities at places where the professor is known. So to maximise your chances, before you choose a professor to ask, have a look at their CV. Find out under whom they did their PhD and with whom they have collaborated in recent years. If those people are still actively involved in some graduate program, your prof might be able to send you there even with holes in your transcript. Just never tell him that this is the reason you chose him...
But all this being said, the harsh truth is, if you truly want to have a career in math, try to transfer to a university with a proper maths program as soon as you can. You'll be able to learn much more at a much faster pace by having more proper courses and by simply having other math students around you.
Also while it might be a prejudice, math courses from some applied university without a proper maths program are generally seen as second rate and lacking in depth. So given the choice between a student who passed introductory math courses for engineers with perfect grades and a student who only got mediocre grades in identically named courses at a place renowned for its math program, many will choose the latter. In the same way, your professors generally won't be the best mathematicians, both in terms of ability as well as in terms of connections, otherwise they would teach at some "better" place.
So I'd suggest to think about transferring again. There might be a scholarship program that you could get into, which would pay for accommodation or a way to condense your physical presence at the other university only to certain days in order to make traveling more feasible. Even if you decide to stay at your university for now, keep looking for opportunities.