We installed Trane zone heat pump system last year but have had nothing but problems with the zone working.
The contractor has informed me today the are going to replace my Trane with a Carrier zone system - which is better system?

They are both the same. It is the install that usually counts. They are both good manufactures.

I am slightly confused wit your post. Carrier's Infnity control zoned system requires an Infinity series system and does not need a bypass damper. I'd be interested if there is another zoning system available from Carrier.

You mention problems, but you didn't define them. It might not be the fault of the zone controller, but rather the ducting.