Dropping out of a master's program to pursue a PhD
When I was an undergrad I had no idea how the machine I was scuttling about in worked.
This was brought to mind by

Is it even typically allowed?

Here's the thing. Masters and PhD students are an asset. Somebody wants you to study with them as much as you want a higher degree. People start PhD programs at all kinds of weird times, even when there's a &quot;normal&quot; intake period. The rules to which you allude are practices that are followed to make administration easy. They are not inviolable Laws.
So my advice is to talk to the academics involved. First, your current advisor, then the other person. While you should take care to avoid being seen as disloyal, those people will eventually find out anyway. So try to get them onside from the start.
It's easy to make assumptions about what is or is not possible, to your detriment. So don't assume. Talk.
And that machine? It's made of people. Good luck.