I am vaguely aware that one of the important parameters of running an AC distribution network is to keep the 3 phase demands balanced; effectively to try to keep zero net current in any given 3 phase transmission line. (am I wrong?)
Sort of yes and no !
The currents in the 3 phase conductors can be different - but will sum to zero when phase is taken into account. Yes they like the currents to be balanced since that also balances loads on each cable and transformer core/winding - but it isn't a requirement for the 3 currents to sum to zero. In extreme, consider a 3 wire 3 phase system and put just one load between two phases - you have one phase conductor with no current flowing, but the sum across all three is still zero.
As John says later, aside from earth leakage the sum of the 3 currents will be zero at any point in time. If you want a non-zero sum then you need to provide a current path outside of your measurement zone and induce a current in that. In practical terms, that will mean earth referencing some point at one end and inducing a current between the equivalent point at the other end.
But given that it's a 3 wire connection, the transformers at each end will mostl likely be connected in delta (so no "middle" point) - and I doubt that the operator will let you earth one of the phases
I suspect you'd need to connect a star connected transformer across the phases at each end to give you a point that you might be allowed to earth - and at one end you could put a transformer secondary (a third transformer, or simply a floating signal generator) between the star point and earth allowing you to inject a current which would be shared equally across the 3 phase lines with the earth making the return path.
An alternative might be to connect a load (resistor) between one phase and earth at one end, and a different phase and earth at the other end - this would also create an unbalanced current. Using two resistors from 2 phases to earth at one end would (if you get the values right) avoid changing the phase-earth voltages in the network - something the operator would probably insist on since unbalancing the phase-earth voltages would increase stress on the insulation of at least one phase conductor.
However you do it, you have to arrange for there to be a current which flows from one end to the other via the earth, and using one or more of the cables as the return path. This current should then be detectable.