I'm going to spell this out. And I'll do it in baby terms and you can claim I'm patronising you, but then at least we are all on the same page, ... you get it and your dignity is intact.
A Mercator projection is a cylindrical map. It is called this because it has a 'pac-man' element. You can travel off the left and magically appear on the right of the map. You can't do this North to South. Hence cylindrical.
There is no 'pac-man' element to an azimuthal map.
You want to take the magic properties of a pac-man map and are asking for the maths to turn that into a non-pac-man map.
So you need to first remove the pacman teleportation by joining the two ends into a cylinder. cylindrial projection, right?
Then you need to take the north pole which stretches all the way around the upper surface of the cylinder and form it into a single point. Now you have a cone.
Now you need to stretch the south pole out so as your cone goes flat into a disc.
Best of luck.