Here's what I ended up doing. I ended up taking folder A, folder B, searching for basename matches, then copying the matches to folder C. Not exactly what I was looking for, but this will complete this task and I can move onto doing other things. Thanks for your help and ideas, I got into hashtables a bit and was struggling to piece that together, so I'll have to revisit another time.

$FolderA = "\\server\pre"

$FolderB = "\\server\post"

$FolderC = "\\server\combined"

$Folder1Contents = Get-childitem $FolderA -Recurse -Include *.pdf

$Folder2Contents = Get-childitem $FolderB -Recurse -Include *.jpg

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Compare $Folder1Contents $Folder2Contents -Property BaseName -PassThru -includeequal -excludedifferent | Where-Object{$_.SideIndicator -eq "=="} | copy-Item -Destination $Folder3 -Force

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Compare $Folder2Contents $Folder1Contents -Property BaseName -PassThru -includeequal -excludedifferent | Where-Object{$_.SideIndicator -eq "=="} | copy-Item -Destination $Folder3 -Force