So, I'm not going to mark this as closed yet, as my issue hasn't been tested on the AMDGPU drivers yet. I won't be able to do that until Monday. Will update then.
Using the AMDGPU driver has not fixed my issue. The monitor still doesn't automatically get a signal sent to it, and when I do send a signal forcefully with xrandr, it comes out as 1280x1440 even though the configuration is for 2560x1440.
I'm beginning to think it is a monitor issue, but again, HDMI works fine with console and Windows PCs outputting 2560x1440. It's just on Linux that I have this issue.
I'd like to point out that at one point, the display switched to it's correct resolution after my PC went to sleep and woke back up. Trying that again hasn't worked though.
I've tried unplugging the monitor's power and the HDMI cable after forcing the signal, but that hasn't worked either.
Using Fedora 25 (instead of my current install of Arch), the display worked fine on DisplayPort. It was also the only display attached to the other PC I was using.
I would just use a different cable, but switching would be a bit of a hassle. The hole for cable management in my desk is ever so slightly too small for DVI and this GPU doesn't have DisplayPort, so my only real option is HDMI.