This is not a solution for getting PUMA to work for our requirements, but I figured that I'd give some closure to future readers.
We ended up using PHP, php-fpm, and nginx.
The same server that could barely handle around 1000 users with rails and PUMA was easily able to handle 3000+.
NOTE: This is not to say that PUMA can't scale, merely that for our situation (A single powerful box running both MYSQL AND the application), php performed adequately.
I'm not sure why this is yet, but I've heard that this is because rails tends to want to scale ACROSS servers (eg. multiple servers), while PHP does better at scaling vertically.
If anyone has any corrections, additional information, or comments, I'd love to hear them.