I always find antennas like this exciting…
In general, they can work in three different ways:
As an endfed with a coil to extend the radiator - a 0.05 lambda counterpoise helps (which can also be the shield of the coaxial cable). Then you have a base impedance that is usually brought up to 50 ohms via a 1:49 to 1:64 Unun.
As a random wire - i.e. a radiator of a specific length that is not in resonance anywhere… then you use a 1:9 Unun… and still need a tuner… a counterpoise often helps here too
Or as a ground plane (or other tuned vertical radiator), which then has radials that are also in resonance.
For all these antennas, the placement of the coil is an issue… I would definitely try using an Unun for your antenna…
I don’t have a separate wire for the counterweight on any of my antennas. I always have 0.05 lambda from the “longest” band (normally 3m because of the 60m band) of the coaxial cable and use the shield. The coaxial cable is then wound directly onto a 140-43 Amidon toroidal core… and then has about 30cm to the BNC connector.
I have a 5.8m wire with 1:9 Unun on a 6m Decathlon mast. It can be tuned with the ATU of the KX2 with an SWR of better than 1.3 on all bands 10 - 30m.
BTW… it was my standard antenna at my trip to Cyprus in December…
Good luck - 73 Armin