I'll try to provide an answer, although it consists of questions and suggestions. The title of this question should probably be "What is the most efficient and practical means for delivering 1 kW power to a tethered drone". There should be a "sweet spot" combination of wire gauge, insulation, and voltage. Perhaps look at test leads typically rated 5 kV, and rather flexible. They often use silicone insulation, which is very flexible.
I agree with comments suggesting 3 phase power at high frequency, and it should be possible to use commonly available ferrite cores and bobbins to make a small three phase transformer. It might be easier to use two separate transformers.
If efficiency is not a major issue, it will probably be best to match the impedance of the cable to the load, so that a 2 kW ground station could deliver 1 kW to the drone, "wasting" 1 kW in the tether. In free air, with the heat distributed along a considerable length, heating should not cause a problem. I would suggest using 1 kV three phase which would result in 500 V phase to phase at the drone, with a current of about 1.2 amps per phase. Use the length of the tether to determine the wire size that will drop 500 V over its length. It might even be possible to use aluminum conductors, although perhaps with some steel strands (or maybe Kevlar) for strength.
You can probably find off-the-shelf 480 VAC (720 VDC) switching supplies or DC-DC converters rated 500 watts, and bypass the input rectifiers and H-bridge, using one phase of the three phase supply.
A quick search for "flexible thin high voltage silicone wire" came up with this supplier, and the smallest available is #30 AWG rated 3 kV, and 1 mm diameter. It has about 0.1 ohm/foot resistance, so for 1000 ft, it's 100 ohms, which is a drop of about 120 volts at 1.2 amps. That's just about 10% for a 1200 volt supply. Weight of the copper is about 0.0003 lb/ft, so for three wires a 1000 ft tether would weigh about 1 pound. You can figure the weight of the silicone insulation knowing the total volume and density of silicone.
Silicone has a density of about 100 lb/ft^3, or 2 Mg/m^3 and a 1 mm diameter wire 1000 m long (for 3 strands) is 78 cm^3 or 0.00275 cubic feet. So weight would be about 0.275 lb or 124 grams.