Hi Mem,
There are videos showing the internal parts and the schematic for a HV module and very likely the one you use has the same circuit inside. So the output wires of the module is connected inside to a series string of HV capacitors that are in parallel with a very high value resistor (to help discharge the capacitors).
The circuit inside is a kind of blocking oscillator (Joule thief).
https://www.youtube.com/watch?v=_Zs8nox4QoA&t=395s and
https://www.youtube.com/watch?v=mTx8pMHo4jI&t=1366s In this latter video the guy shows the input waveform of the HV transformer
https://youtu.be/mTx8pMHo4jI?t=1602 The 1st video is 8 minute long only and it questions logically the 400 kV claimed HV output for such modules at the start of the video. I mention the internal circuit to get as much information on the HV module as possible. So the available HV may bein the range of 20-30 kV or so (depends also on input voltage of course).
Regarding the input current dropping to about its half value when loading the output of the module, it may be caused by the reduction of core saturation in the HV transformer (due to the secondary coil's counter current), hence the primary coil's inductance (so its AC impedance) can increase. See such a HV transformer either in the above videos or here
https://www.ebay.com/itm/401543244152 to consider the very thick input wire used, this is why I mention possible core saturation.
Another cause for input current reduction may be that the spark gap arcing creates negative resistance for your primary air core coil and this may make extra oscillations thus easing the load on the HV module. The latter may not happen like that but the negative resistance can be true if the HV voltage level remains in the range where the negative resistance happens in the arcing spark gap.
Yes, the spark gap can affect the reading of meters especially the digital ones due to the possible high EM field radiation. Try to use very short connections to the meter to reduce EM field pick up and also use some hundred nF capacitors directly across the 3V input of the HV module and also across the 2 series batteries, to short out the unwanted EM pick ups. If you happen to have an analog Ampermeter that has moving iron or coil only and has no other active circuit, that may also help, connected also with short pieces of wires between the module input and the battery.
Because such HV modules may have input current draw at 1 or 2 Amper peaks, here is a 3A max step down converter
https://www.ebay.com/itm/272795588317 to consider for looping.
It has 3.3V DC output and with a series diode you can reduce it to 3V or little below for the HV module input. The series diode helps isolate the 3V battery from the converter 3.3V output when looping, till you remove one of the batteries to see whether the circuit is able to self run...
Regarding tuning your coils, the 'how to' is mentioned nicely by member nix85 above. What I would mention is that the spark gap when fires intermittently like a switch, this process inherently creates flyback (kick back) pulses across your big air core coil. And there is already an inherent resonant frequency for your big coil which comes from its self inductance and its own self capacitance but the resonant frequency of such primary coil (high L with low C value) can be much higher than the repetition frequency of your spark gap. You can adjust this latter by the gap distance of course and also by varying the 3 V DC voltage input to the HV module to get a varying HV ouput from it.
Here is step down regulator with variable output between 2 V and 34 V, also with very small idle current, max output current is 3 A:
https://www.ebay.com/itm/404234781488 Gyula