It sounds like the paths to the music files are different on each PC. You can check this by opening the playlist file in any text editor (e.g. Notepad++) and checking the folder paths. If they are not correct for your computer A then MusicBee will not find them and I don't think there is a way to link to all the files in a playlist in one go.
Other than making sure that the full path on both computers is identical, the easiest way to do this would be to edit the playlist file before importing into your computer A (using a replace tool to select the wrong part of the path and replace it with the correct path).
(Could a Mod move this to questions)