Will adding blown-in insulation to my floor cause a moisture problem?
Should not cause a moisture problem. Cellulose has interesting and useful properties with regards to moisture movement on the one hand, and the vapor barrier is still going to be on the warm side unless there's a lot of space.

Essentially, cellulose has been tested multiple times to have such low air movement through it that it effectively does not need a vapor barrier.

Blowing it in without tearing the Kraft face may require a good deal of care.