I would first experiment with small traditional transformers.
I was able saturate this bigger (choke I guess) E trafo core, while I've one coil was in series with those leds above (anti-parallel configured to detect AC, not DC only), however problem with those E cores is that coils a in the middle and we can only try expose this thing to external magnetic field from both sides, which means to be able apply symetric external field for each half of AC waveform, we have to put two magnets from opposite sides and.. we have a problem-magnetic field in the middle of this E core will cancel if we use the same magnets, so only part of the coil will see strong magnetic field.
In the case of, when those two magamp coils are not in the middle, when external field is apllied in the middle (by magnets or DC voltage) than magnetic flux is divided and flows to each side of magamp, so when additional diodes are present it makes symetric saturation easy at the same level of additional magnetic flux in whole AC period, so it looks like there is advantage when those (transformer) coils are not in the middle of E core.
Anyway this choke has much higher inductance when not saturated, while this simple ferrite bead one side has probably less than 100uH, since it blocked 100kHZ AC at similar way like this known SMD 0.1 Ohm 100uH inductor at 100kHz.
So, yep this choke (each coil has) much higher inductance than SMD inductor and ferrite bead tested, but magnetic flux inside this thing is asymetric (the same single SMD inductor), when applied strong magnetic filed to ONLY one side of this transformer (SMD inductor), so probably we should be able to see that one of those LEds make more bright light, since those inductors will saturate asymetrically when we'll be close to its maximum magnetic flux density (saturation)
Low power magamp controled by strong magnet looks quite interesting, however we have to know where apply external magnetic field, since in one direction it will be added to internal magetic flux created by its coils, but otherwise they will try to cancel each other, so we can still get this magamp core saturation effect when magnet field will be very strong, but when those magnetic fields are in correct direction, then the same effect can be achived using smaller magnet.
It was interesting experiment-I need to order a few stronger neodymium magnets to experiment more with this thing, but early results are very good and this thing can be usable even with ferrite bead