I actually got this working after stopping the Docker-Traefik and just running the other Traefik on the right ports (80/443). I think it was getting hung up at the ACME part or something (I have no idea how the internals work), but it appears that the internals were not fully started, and so it was returning 404 for everything. Once it was running on the correct ports, everything behaved as expected.
This issue was what tipped me off: Getting 401 or 404 HTTP errors for every service