$5,000 to $7,000 sounds about right for a new NG furnace. My parents just replaced theirs and it was about $10,000 installed for a 97% efficient furnace for their 1000 sq ft home
You would likely spend a similar amount for a heat pump that would go in the same space.
How large are the temperature swings in your home. You said you turn off the heat at night - how cold does it get. What is the temp you heat your home to during the day when you are there?
Heat pumps can vary temperature fine but they are slower to heat up a cold room than a gas furnace because for a ducted system a furnace puts out air at 150F to 160F while a heat pump is 105 - 115F
My heat pump is set on a program:
- Mon - Thur 62F at (9 pm), warm to 67 in the morning (4:30 am), drop to 65F at 6:00am, heat to 72F at 3:00 pm, then 65F at 5 pm.
- Friday we are home so it is 69 from 6 am to 3 pm
- Sat / Sunday it is 69 from 6am to 9 pm (no peak electrical rates on weekends)
We are shifting our temperature a max of 10 F which works fine. If you are heating from 45F in the morning to 65 - 70F and then turning off the heat again when you go to work then on again at night when you are home I doubt a heat pump will work well for you.
With a heat pump as a primary heat source you would also need back-up heat for at night when the outside temp drops below the heat pump rating. (assuming you heat at night) A small electric heater would do that.
DIY you would likely need to go with a 36,000 BTU ductless in your kitchen / living room and then a smaller 24,000 BTU unit on the other end of the house in the master bedroom. That would likely cost as much as a furnace or singled ducted heat pump
Realistically just replacing the furnace is likely your best bet in your climate.