Hi Chris FFullDeckOfFools ,
If I understand it correctly, what you want is not possible.
You could try using custom 'radial UVs', like this for instance:
That 'compresses' the Texture near the center, so will probably not work for you, but like I said, you seem to want the impossible.
Found the solution here: https://blenderartists.org/t/uv-unwrapping-unique-shapes-seamlessly/1517531
It actually wasn't an issue with my UV but instead the texture itself. I edited it to be seamless from left to top of texture as well as left to right and vise versa in a photo editing program.
Just in case someday you need a randomizing behavior, there's Scattershot:
https://cgcookie.com/downloads/scattershot-procedural-image-scattering-addon