I don't know if it's possible to compress on the way in using BIAB. But, if you have to use a high compression ratio, say, 4:1 and higher to control the singer it is going to start sounding squashed and unnatural. A low ratio, 2:1 or so will probably not mess up the take but is not likely to tame the input VU meter enough either.
If it were me, I would have the singer review some microphone techniques and practice with the microphone and headphones. Then redo the take/s. He'she may even appreciate the improvements he/she makes as a singer.
Also, it's okay to record with quite low input levels. Just so long as the noise in the room is also very low.


Does the noise in your head bother me ?