I do not know the name for the technique I am looking for, but I am essentially trying to take this map: And have it distorted into a mercator projection to complete this map: A program that allows you to set the center point and radius then distorts it into a rectangle would be enough to do the job, however without knowing the name of this technique, I can't finish these maps.
The Mercator projection is fundamentally cylindrical: it first maps the globe onto a hollow cylinder (so that the South Pole is either not represented or mapped to the entire bottom edge), then unrolls that cylinder into a rectangle.
While there are various other steps in the Mercator projection, what you want here is a polar-to-rectangular conversion: mapping the polar coordinates (radius and angle) to the X and Y axes. And Photoshop conveniently has that built in, as the "Distort > Polar Coordinates" filter. Position the South Pole at the center of your image and choose "Polar to Rectangular", then rotate and scale as necessary.
If you visit the Wikipedia page on Mercator projection you will find
The Mercator projection is a cylindrical map projection
The spherical approximation of Earth with radius a can be modelled by a smaller sphere of radius R, called the globe in this section. The globe determines the scale of the map. The various cylindrical projections specify how the geographic detail is transferred from the globe to a cylinder tangential to it at the equator. The cylinder is then unrolled to give the planar map. The fraction R/a is called the representative fraction (RF) or the principal scale of the projection.
I hope this can help.