Small Cheap Indoor Natural Pool - what is the cheapest filter pool option?

I am wanting make a cheap indoor natural pool filter for a very cheap second-hand above-ground pool.  

I am scared that the cost of this is going to way out-strip the cost of the pool.

We live in a very temperate climate.   average min/max in mid summer is 13C/22C  Average min/max in mid winter is 2C/13C.  Record low is -4C.
The system will not freeze, but it will need heating pretty much year round to reach 26C swimming temperature. I would want both pool and pond water heated and the water in a closed system.

I am looking at an area of 4-8M2 (40-80 ft2) and a volume of 3000-6000L (740 to 1500 Gallons) to start with.

I have 8 acres and and a fairly secure rain/bore supply, and don't eat fish, so have not really dabbled with aquaponics much.  I have grown duckweed for chicken food, in shallow ponds, however, and would like to do so again.

I have seen duckweed mentioned as one plant with filtering roots, and am wondering if it could be the only one used.  Could I pump up to a 44 Gallon drum filled with sand that empties into a shallow pond above the pool filled with duck weed and then into the pool itself?

Also, I am not sure what keeps the plants living in the natural pool pond, as I am guessing the last think you want to be doing is dousing them with your manure tea or other natural fertilisers!  Do we throw off enough dead skin etc to keep fertilising them?

I like your idea regarding using duckweed.  I found some threads that might be of interest:

https://permies.com/t/93029/newbie-aquaponics-filter

https://permies.com/t/4468/pond-biofilter

https://permies.com/t/4936/pool-filter-pump

https://permies.com/t/144394/Grow-Hydroponic-Plants

https://permies.com/t/152807/Natural-Pool-Integrated-Wicking-Beds

https://permies.com/t/151988/Organic-pool-subtropics-identifying-plants

I think the sand filter is the way to go.
Its cheap,  simple and easy to add more drums to in series or parallel.
I would want to grow willow or cattails in the barrels themselves,  but I've never seen it done.
Azolla fern could feed itself and can be fed to poultry.

What kind of building will house this pool?
I'm guessing greenhouse or hoophouse?

this is what i've been building the last two weeks. 1st step was to rebuild the green house with Bi-wall polycarbonate panels. 2nd was to build the frame for the pond. its 9' 8" long 5 ' wide and 38" deep. the frame is made using recycled wood. some left over from other projects. the frame was then lined with 1/2" ply and then lined with 1" insulation. a pond lining was then inserted and secured with trim wood along the top edge. the lower half of the green house also has 2" insulation. Filtration system is made from 2 30 gallon drums. 1st drum is a swirl filter and the second is a bio filter.
the plumbing isn't completed yet for the grow tubes. and i still need to build the flood beds which will be on the other side of the green house. I also built a solid wall on the north side and insulated it with 3" insulation as this side has no benefit for sun and gets hit pretty hard by the winds in the winter.
Next part of the project is a solar heater for the pond. i'll update pictures when its done.

added the solar water heater. this will only run when the water temp drops below 65 and will switch back off if it goes above 75. i need to figure out how to keep it from running at night.
so this is made from a 4x8 sheet of ply. 3 2x4x8 lumber , 4x8 insulation and a 4x8 bi-wall polycarbonate sheet.
one question i have is that the sheet of insulation has a reflective film. is it better to have this or a black covering?

This could be a cheap option. Pumping air.

https://m.youtube.com/watch?v=RcB5ZP1GuOA