Perhaps your ductwork is leaking or there is another easy fix.

Cost it out if you have duct work new central air may be cheaper to install then minisplits.

We bought special furnace filters, cleaned the duct work, ripped out all the carpet, cleaned and sealed the wood underneath, and replaced the carpet.  

And depending on what you want it will run into a small air handler with a coil and fan, or if you choose a “mini split” there will be multiple line sets running to the “heads” basically the fan and coil all in one that mounts inside the room and looks very nice.Depending on the square footage and what the original heat pump was (air handler with duct work already installed?)

Remember to also include cleaning the heating duct work if you have forced air in the house.

When the ductwork was installed, did they move or remove some of the lolly columns?  

All new fixtures. 5000 Replace improper basement ductwork 3225 A/C coil and condenser 3200 Stairway into basement - bring up to code 2250 Washer/Gas Dryer (stacked) 1800 Paint throughout 1500 Transfer taxes 1200 New carpet - stairs and both bedrooms 1000 Gas water heater 800 new fridge 800 Exterior paint 800 Demo/creation of fridge location 725 Replace wood rot along roof line and cap with metal 700 Roof coating 700 new gas stove 600 Trim repair, doors for both closets, door for 2nd bedroom 500 Window blinds throughout 300 Minor siding repair - rear 200 New screens 150 Total costs 34200 Offer ALL CASH 21799

OdorXit.com has a water and essential oils solution that as long as you get it to ALL the urine - drywall, concrete, the 2x4's behind the drywall, even the duct work, it will eliminate the smell 100%. 

We had heat pumps with duct work north of Toronto Canada.

You could leave the asbestos in-place until such time as it is convenient for you to remove it, provided it remains undisturbed and there is no path by which friable asbestos can enter the living area {essentially a good vapour barrier around all fixtures and properly sealed ductwork if it passes through the attic}.