I'm not an expert when it comes to Proton and I don't own Rime so I don't know how much I am able to help. Two lines in your error log seem interesting to me:
Proton: Missing or invalid openvrpaths.vrpath file! [Errno 2] No such file or directory: '/home/<ComputerUsernameCensored>/.config/openvr/openvrpaths.vrpath'
and
wine: /media/<ComputerUsernameCensored>/DATA/Programs/steam/steamapps/compatdata/493200/pfx is not owned by you

According to it's Steam page Rime doesn't seen to be a VR game so the first line confuses me a bit.
The second line looks like permission problems. You should check the directory and make sure you have permissions to it. You could also try deleting the pfx directory, just make sure to back up save and config files first if you don't want to lose them.

The 2nd line looks like the game is on another disc - if it's your window drive it needs to be mounted by your user not root ( or you need to tinker with mount options to give the correct permission ).

Alternativly just install it on linux and spare your self the hassle of having to deal with NTFS

Thanks for all the help! I finished playing in windows, but that might fix some of my other games, since they're all on an ntfs disk.