Well, I guess I get to answer my own question, for once!
The key to make the context menu appear in Theater View is the special "Application Menu" key. I did not find it at first because the wireless keyboard I use with my HTPC does not have this key!
After poking around the web, I finally found the KeyCode to use with the Windows Message Command to "send" the Menu key to MC12 from Girder:
wParm is 27000 (The MediaCenter command for "send a keystroke"). The lParm is 57437 (0xe05d), which is the scan code for the Menu key.
For reference, I found the key codes for "special" keys here:
http://www.win.tue.nl/~aeb/linux/kbd/scancodes-6.html . I also found key codes for some of the "normal" keys here:
http://flash-creations.com/notes/asclass_key.php . I have not yet found a single reference for all key codes. For example, I can't figure out how to send "Alt-Right Arrow" (although I have worked around the need to do so in my setup).
I hope some others may find this information useful.
-- Soundman