Thanks, I've heard of DaVinci Resolve and found out shortly after commenting that there is a free version. We should give it a try.

I've got ffmpeg and Handbrake for transcoding but for some reason they both caused issues in Premiere Pro still (audio sync, choppy/repeating footage, etc) on those files. I'm not very experienced, so that might be on me, but it didn't seem to happen outside of Premiere Pro.