So, you have a conventional gas furnace, evaporator coil, and air conditioner system now, as I understand. You were quoted for a "heat pump and air exchanger" - my question is does this mean getting rid of the gas altogether and installing an air "handler" and heat pump, or just installing a heat pump and a new evaporator coil to go along with the furnace. FYI, an air handler has a fan and an evaporator coil, and no furnace is used along with it. I'm just not clear on what the replacement plan is--get rid of gas and go straight electric, or keep the gas and just add-on a heat pump (which would essentially mean the gas heat is never needed except for very rare occasions, and only provides the airflow).
One more thought...if the compressor burned out, I'd see about replacing the copper lineset that goes from the outdoor unit to the indoor unit, if easily done. Otherwise other measures would need to be taken to ensure contamination is removed from the system.