@MissAussie Please take a look at the following pages first (for complete details):
<a href="https://www.homeaffairs.gov.au/trav/visa/visa-about-to-expire-or-expired/extend-my-stay">Visa Extension</a>
<a href="https://www.homeaffairs.gov.au/about/corporate/information/fact-sheets/52b-nfc">No Further Stay Fact Sheet</a>
According to these pages, if you have a "No Further Stay" condition under your current visa, you can't lodge for a student visa onshore. I believe you have to leave Australia and lodge the student visa off-shore (you can't send your application to Australian embassy/consulates outside Australia while you're still in Australia).
<blockquote class="Quote">If you have a 'No further stay' condition on your visa, you cannot apply for another visa while in Australia (except a protection visa or a temporary visa of a specified kind). You must depart Australia no later than on the date on which your visa expires.</blockquote>
<blockquote class="Quote"><b class="Bold">If your visa will soon expire</b>
Requests for waiver of 'No Further Stay' conditions cannot be decided 'on the spot' as they require careful consideration against legal criteria. It might not be possible for the Department to make a decision on your request before your visa expires. Lodging a request for waiver of a 'No Further Stay' condition does not automatically mean you will be in Australia lawfully until the request is decided. If your visa expires before the Department can decide your request, you will become unlawful. This has serious consequences.</blockquote>
<blockquote class="Quote">A 'No Further Stay' condition is a condition that prevents the visa holder from applying for many temporary and permanent visas while they are in Australia. 'No Further Stay' conditions include 8503, 8534 and 8535.
If one of these conditions has been imposed on your visa, you cannot apply for another visa (except a protection visa or a temporary visa of a specified kind, while you are in Australia).
If you don't have the right to remain in Australia, you must depart.
When you depart Australia, a 'No Further Stay' condition will not prevent you from applying for other visas.</blockquote>
The very first part of the student visa application will already ask you where you are applying from and why you are there (if you live outside your home country, you have to tell them what kind of visa do you hold). So from here palang malalaman na ng DHA if eligible ka to apply for student visa or not.
Have you secured your eCoE and other student visa application requirements? In my opinion (FYI I'm no migration expert/agent ha), it might be best to return to Manila first and apply from there para hindi ma-reject yung student visa application (sayang yung visa fee), kasi clear sa website nila na hindi ka pwede mag-apply while you're onshore. Kung 3 weeks away pa ang departure mo from Australia and July intake ka, may risk na hindi umabot ang student visa mo (in time for your course start). Maybe you can consider delaying to the next intake para hindi ka gahulin sa oras for your student visa application?