Sorry, I re-edit my post after realizing that you're living in a temperate country where the air inside your house can be cold during winter.
But if you have heaters to heat up the air inside your house, probably it should be fine for ideas 1), 2) and 3).
But if your house doesn't have heaters to heat up the air, then it's better to have a glass or acrylic sheet that can cover up the top of the tank.(idea 4).
If I'm not wrong, the air temperature above the surface of the water shouldn't be too much different from the water temperature during the winter for fish that need to breathe from the surface of the water(Bettas, Gouramis).
1)If you can keep the water level at around 8cm below the rim, probably you don't need to have a hood.
2)For cheapest method, you can get some thicker plastic sheets that are used for craft works but may need to put a weight on top.
3)Another method is to use a net/plastic mesh to cover.
4)Lastly, if you can get some acrylic sheets or a glass top from the fish shops will be the best.
Some shops may carry the plastic hood but usually they are only available in some standards sizes and they are not cheap.
For filtration, a slow flow rate hang on filter or a sponge filter are ok.
For hang on filter, probably a flow rate of about 3-4 times the volume of the tank.
Example: For a 20 gallon tank, flow rate: 60-80 gallons per hour.(240-320 liter per hour).
By the way, make sure that the Betta doesn't carry any disease.
It's difficult to treat disease and it may cost you more to buy the medications.