So both responses have merit.
ArcGIS Pro is NOT recognizing the Windows folder options (Show hidden files, folders, or drives) so via Asrujit SenGupta's answer, you can just get around that by dumping the path in and going from there.
Dan Patterson's answer is correct in that if I were to actually unhide the folder (rather than show hidden files, folders, or drives) it would then be visible in ArcGIS Pro's folder selections. However, seeing as AppData is one of those folders that hidden by Windows on purpose I don't necessarily want to unhide it (not that it matters really - no one else is on my computer).
It would appear to be a backward step. Sure there are workarounds but why not go back to following the Windows folder options settings?
So with these settings:
This is what ArcMap shows:
But this is what ArcGIS Pro shows: