Generally it's a good practice not to mix light sources. The CC filters or Fld/Flb filters will work for you provided the light/film combination is not mixed. With B&W it's no problem.
Stay away from mixing with sunlight as best you can when using any of the above filters. Can you draw the blinds to exclude natural light?
Frequently when a serious pro shoots they will actually filter the surrounding natural light sources to balance to the flourescents. This is done with large Gels taped over the windows. This way the photographer has only one light source to filter for.
Must be nice to charge the client, eh?