Instead of using a transition piece where the two floors meet, pull the carpet tight and wrap the edge underneath. This will hide the unattractive cut edge of the carpeting. You can use staples or whatever method your professional installer recommends to hold the carpet down.
Slide the edge of your vinyl floors underneath the curved edge of the carpeting to create a seamless transition.
Sand the bottom of the t-molding track and lay down a strip of glue at the edge of the carpet. Place the track so that the edge of it slides underneath the tack strip at the edge of your carpeting. You want the edge of the carpeting to go over the edge of the track.
Install your vinyl floors up to the other edge of the track and snap in the t-molding, hiding the edge of both the carpet and the vinyl.
Install your vinyl floors up to the carpeting until you reach a distance that can be covered by the end cap (roughly 1/2 - 3/4"). Sand the base of the end cap, lay a strip of glue, and press the end cap in place.
Wait for the glue to dry, then tuck in the edge of the carpeting and trim off any stray fibers with a utility knife.