My name's Kel.
The least expensive approach I can think of is having the dog stay in one room and closing off the return air register so the smell from the dog doesn't get pulled into the ventilation air stream.
You could also weatherstrip the door that connects the spaces. That would include the perimeter of the frame and the gap between the floor and the bottom of the door.
If the house has carpets have them steam cleaned with NO chemicals.
Same for any upholstery.
A more expensive solution is to add an electrostatic scrubber to the furnace. This will remove particles that the smell binds to. Google HVAC contractors for your area. Get estimates from three.
What other questions do you have about the problem?