When you double click a file in the Finder (or a Mail attachment or whatever), the Operating System receives an Open FIle Apple Event message. It attempts to work out which application "owns" files with that extension, launches the application, and forwards the Open File event message to it.
Unfortunately, the Civ4 game executables have not been written to respond to Apple Events such as Open File, so although the OS can recognise that a .Civ4BeyondSwordSave should be opened using BtS, and it can launch BtS, the Apple Event falls on deaf ears.
I have not played PBEM games, but it is my guess that BtS doesn't know how to send the correct Apple Events to Mail, either. This would be needed to persuade Mail to create and send an email when you select that option.
So I think you are probably stuck with doing it manually as you are currently. Note that there are some shortcuts that may help:
You can put a shortcut to your ~/Documents/BtS/Saves/pbem/ folder in the Finder's left hand panel, or in the Dock at the right hand end. Drag and drop incoming Mail attachments onto that folder to put them where they need to be to open with the Load a Game menu option. When you have saved after your turn, just click that folder shortcut to show the list of saves, and then drag the new save onto your Mail icon in the Dock. That will open a new Mail message with the file already attached.