I would sing the vocal in BB. Then add the harmonies with the TC in BB. Just set them so they each save to their own track.

If you sing into BB, you will not have to mess with sync'n em up. Write harmonies to the whole vocal track, much easier to cut out the parts later then to try and write them several times over snippets of a vocal track.

Import them into RB, cut parts, set FX and mix it down.

Trax