Forget tiff files. Use png--you won't have to make an alpha channel.
Cut your object out and put it over a transparent background in Photoshop (the checkerboard). Don't bother making an alpha channel--just save it as a .png. Import into Avid with Invert Alpha and you should be good to go. Avid will interpret the checkerboard "transparency" as your alpha, and it should do it without the white edge.
This has always taken care of the white "ghosting" from other formats that require you to create an alpha channel (tiffs, targas, etc...).