I reckon the Soap reactor contains circuitry such that each 4.5V 30mA output acts just like a battery without the need for the extra resistor. The no-load voltage is 4.5V but more importantly it will provide a current up to 30mA to illuminate the LEDs; as it provides this current the output voltage at the soap box socket will fall a little but not so much that it becomes less than the forward voltage requirement of the LED.
I will send you some LEDs to experiment with to confirm my thinking. If they are damaged then it does not matter but we don't want to damage those in the figurines.
I will draw out and explain the experiments in a later email - may not be today though.
When I look at the youtube explanations for the soap reactor the lead from the reactor turns into two thin wires at the end of which is a stubby cylindrical shape which substitutes for one or more 'hearing aid batteries' - this is inserted in lieu of the battery/batteries. Is this what you have pictured with the meter prongs at either end?
Yes that is what I have pictured with the prongs at either end.
I do have some spare LEDs I can play with here. I have some these if they will work:
Broadcom3.6 V Green LED 5mm Through Hole, HLMP-CM2H-130DD | RS
uk.rs-online.com
Vishay3 V Green LED 3mm Through Hole, TLHG4401 | RS
uk.rs-online.com
Vishay3 V Green LED 3mm Through Hole, TLHG4900 | RS
uk.rs-online.com