The alternative launcher (replacement for "Monkey.exe") won't help here. "Monkey.exe" and the alternative launcher are just wrappers for the actual game executable "Monkey4.exe" in the same directory. You can actually launch "Monkey4.exe" directly using the "Run Command..." in Crossover. You can pass the following options on the command line:
-gl use opengl rendering (seems to default to directx). (Required option, see below).
-w run the game in windowed mode (an option not available with the original launcher!)
-16 set 16 bit colour depth
Note: I found the executable immediately crashed if I didn't set the "-gl" option. You might see an error on quitting the game also, even with this option set. I don't recommend running the game executable directly, unless you specifically need to run in windowed mode.
The cutscenes are in bink format. The bink files appear to be embedded in the ".m4b" files in the Movies directory, instead of being in separate ".bik" files. The game executable plays the embedded bink files using the DLL binkw32.dll. An incompatibility of this DLL with Crossover with respect of the video rendering routines is likely the cause of the black screens. (I've been unable to find an updated version of this DLL).
I have no working solution for playing the cut scences in game, but I have found a (convoluted) workaround for viewing the cut scenes outside of the game.
To play a cut scene open (a copy of) an ".m4b" file from the Movies directory with a Hex editor and strip the start of the file (should begin with "SMUSH") before the bik file header "BIK". Save the resultant file with the extension ".bik". The contents of the file should now begin with "BIK". ENSURE YOU SAVE TO A COPY of the original file to avoid overwriting the original! You should now be able to play the resulting file with either Radtools through Crossover or the BinkCarbonPlayer, both available from Bink Downloads.
It should also be possible to write a wrapper script or wrapper application to play the original cut scene files directly with radtools or BinkCarbonPlayer. I leave this as an exercise for the reader.