How valuable would it be to be able to do clamp on testing?
Is this a one-off situation where you are only doing one test, and using the standard (but slower and more expensive) fall of potential testing makes sense, or are you developing a regular testing program where coming up with a specialized tool might make sense?
The reason is that in principal you _can_ clamp to the wires feeding the ground rod, but only if you do it correctly, and it is possible to design an instrument which would work with multiple wires coming off a single ground rod. This might be worth it if the engineering cost of designing the test ends up saving money over doing lots of tests.
@synchro already answered that if there is a single wire going to the rod, you can just clamp around that and get a useful measurement. Additionally, if you have multiple wires but can clamp to them all at once you would get a useful measurement.
The key is to understand how the test works. For a nice explanation, see page 10 and 11 of:
https://www.instrumart.com/assets/Megger-Clamp-On-Guide.pdf
The clamp on meter is _not_ measuring the resistance of a single ground rod. Instead you clamp it around a conductor and it measures the resistance of whatever circuit that conductor is a part of.
If you clamp on to a single ground rod, that circuit consists of the single ground rod placed in series with the set of all the other ground electrodes in parallel, plus the various interconnecting wires. This is the ideal case for the measurement. The single ground rod resistance is much larger than all the other resistances in the circuit, and 'dominates' the result.
The clamp need not be large; the meter can be separate from the clamp, and there is no physics requirement that the clamp have big handles hanging off the side. This is simply how the tools are designed and built.
If you clamp on to a single wire feeding a single ground rod, then you have the same topology and essentially get the same measurement, because you can pretty much ignore the resistance of the wire in this case.
But if the ground rods are arranged on a loop of wire, and you clamp somewhere on the loop, then you end up measuring the resistance of that wire loop, basically ignoring the ground rods.
If you 'pinch off' a section of loop, and have the clamp around both sections of wire feeding a single electrode, then again you have just 1 electrode on one side of the clamp and all the others on the other side of the clamp. The two 'loop' wires cancel out so that you have no net interaction with the wire loop, and again you are back to looking at the single electrode separated from all the others.
So if you can get _all_ of the wires heading to a rod into your testing clamp, you will get a good measurement. Your diagram shows ground rods in sort of a triangle arrangement. If there are locations where _all_ of the wires connected to a single rod can be clamped, then just clamp there.
The picture you attached showed 4 wires welded to a single rod, two going left and two going right; no way to get a clamp around that. But it would be _possible_ to put two clamps around the two sets of wire. If these two clamps are arranged correctly, the _sum_ of the two clamps would cancel out and you would be left looking at the single rod. I don't think any manufacturer sells and instrument that works this way, so going down this route means something at least semi-custom. For high frequency (RF) testing applications, current measuring probes and 'injection probes' are available off the shelf, but I don't know if they are available down to 60Hz.
My advice: do the FOP test and move on. Add to the specification for future jobs that the conductors be routed to permit clamp on testing. But if you are setting up a testing program that will run for years and cost more millions, then explore getting a custom instrument which could make measurements on two (or more) sets of wire coming off a single ground rod.
-Jon