If it were possible, I would like to be able to hit the Audition button and have Cubase automatically start to record in sync with the Montage. Or, alternatively, I would like to be able to copy the MIDI files for the Auditions directly out of the Montage to the flash drive.

Is either of these two ways possible?

No.

Is there another way?

Yes. Recall the Performance.
Make note of the Tempo.

Setup Cubase to record MIDI
Set the Tempo to match the Performance.

Start Record in Cubase
Then once the transport is running press the [AUDITION] button to begin the Audition Phrase.

Once the data is transferred you are going to *Select All* and drag the events to the nearest Bar line.

Details:
__ You want the Audition MIDI data to travel Out via MIDI, but in this case, you don’t need to echo it back to the MONTAGE, nor do you need to disable Local Control...
__ Set your MIDI Track(s) to receive MIDI from the MONTAGE. There is no need to set a MIDI OUT to return the data (during record).
__ Avoid using Quantize in this operation.
__ We are not going to use MIDI clock, simply set Cubase to the Performance Tempo, we will manually align the data.
__ All MIDI data will be recorded to the Track — you can divide it by channel afterwards or you can use the MONTAGE Multi Channel Record template (recommended for flexibility in analysis)… this creates a MIDI Track exclusively for each Part and a separate Sysex Track.
__ The number of Parts will dictate the number of Channels of data will be sent by the Audition Phrase.
If you are using the MONTAGE Multi channel Record template… set the Sysex Track to record your Super Knob and Scene events, and activate the same number of Tracks as there are Performance Parts.

If your “Super Knob CC” = Off, the Super Knob will send Sysex.
If your “Super Knob CC” = a #, that will be recorded on Channel 1
If your “Scene CC” = Off, the [SCENE] events will appear as Sysex.
If your “Scene CC” = a #, the [SCENE] events will appear on Channel 1

The advantage to recording the Sysex to its own Track, is you can Mute it easily. Or you can leave it active while you Solo a music Track, allowing you to hear how any single Part is responding to the Super Knob. Each Track can have an entirely different reaction to Super Knob movement.

Cubase skills: Select All events and drag the group to the nearest bar line. You can do so by aligning the first note-on event to the nearest measure line in the Key Editor.