well, my guess is too much gain on the darlington along with phase shifts causes instability in the control loop. maybe lowering the high frequency gain by connecting a 2.2n (or maybe bigger) across the force and sense pins could help.
The output bypass capacitor, C2, needs to go to the same ground that the input bypass capacitor, C1, goes to and not the operational amplifier output. The simplified LTC6655 schematic shows a common source output stage which is going to be picky about decoupling.
Adding a base-emitter shunt resistor or pulling the LTC6655 force pin down may be needed to force the output stage into class A and prevent the sink output transistor from becoming active because the LTC6655 ground is not going to be low impedance with the operational amplifier driving it. Even that may not work. The circuit may need to be changed so the operational amplifier is driving a low impedance capacitance to decouple the LTC6655 ground. In the video I suspect that is where the low frequency oscillation with the larger capacitor originated.
I would decouple the collector of the transistor separately from the reference Vin and isolate them somewhat. This would not be the first time I have seen the simple current gain stage couple back into the reference supply causing stability problems. The relatively high impedance power and ground connections of the solderless breadboard are going to exasperate this issue.
I would be surprised if SPICE models the above problems well without doing enough work understanding the circuit to not need to use SPICE and I see no LTC6655 macromodel anyway.
It may be simpler to use a more traditional precision current source design with a separate reference, error amplifier, and single ended output stage. The LTC6655 is going to be finicky because of its common source output stage.