Correct. Air exchanger is for IAQ. It sounds like he is talking about make-up-air, which is a different thing. However, a heat pump should not need make up air, as it doesn't use combustion air.

Make-up-air may be required if you are depressurizing the house with exhaust fans, clothes dryers, or the like.

Perhaps your ducts aren't up to snuff. That seems to be a common HVAC issue in new construction these days. Was a Manual D duct calculation done, along with the Manual J?

Here is a link that might be useful: Make Up Air Controller

And make-up air shouldn't affect one room over the other--it should reduce airflow relatively evenly.

The room with the most difference is pretty far from the air handler. Both manual J and D performed. The way it was explained to me is that because the house is so well insulated, the air exchanger/make up air would increase the interior volume of available air. We are really confused. With three kids, the doors are always being opened and I would imagine plenty of fresh air is constantly being introduced.

Isufan:
air exchangers will NOT be a remedy to increase airflow as one of the features of them is to exchange inside air for outside air in a balanced fashion. If the system is still accessible, which I'm betting it isn't, were balancing dampers installed? Possibly adjustments there could improve the situation. If not accessible, you're pretty much left to adjusting the dampers in the actual supply diffusers upstream of the problem area. Oftentimes in residential systems, unvaned trunkline offsets, elbows and size transitions can cause turbulent areas in the duct wherein the takeoff location can allow a significant amount of air to bypass it. I had one system installed where a supply outlet was actually at a negative pressure! Freaky! Luckily, I caught this before the lower level had been drywalled and was able to relocate the takeoff to solve the problem.
Air exchangers can be an expense and to remedy this I think a pointless one. Unless your contractor is willing to allow you to crucify him when it doesn't work!
Have him try to balance it, or maybe add an additional supply opening in the problem areas if possible. Good luck!

LSU -- I asked our contractor and HVAC guy about a fresh-air exchanger (my father-in-law's contractor told him it was necessary with foam insulation). They both said it wasn't because, first, we have electric heat, and, second, you're going to open an exterior door at least twice/day. We have 3000 sq feet, all foam envelope, and three HVAC units.

I was in my father-in-law's attic one day, and the air exchanger that his HVAC guy put in was actually closed off with a piece of plastic shoved into the intake side!

Geaux Tigers!