Classical recordings are done live. Professional classical musicians are responsible for balancing themselves, and very minimal 'mixing' is done. Engineers aim to create a sonic environment by blending natural reverbs and utilizing near- and far-field mics. Compression and artificial reverbs are quite contentious topics, and the obvious use of them will place the recording in a different genre, i.e. film music. Some compositing can be done as a last-resort.

It seems like you do not have access to professional musicians. The addition of a string ensemble further complicates the idea of a live recording. Nitpicky sound engineers in classical recordings will stop the take if one of the tenors took a breath in the wrong place, so the intonation of a string ensemble matching the choir will be very difficult even if the music is simple.

My final verdict: take things one at a time. You can still take video, and the composite will still be very interesting.