Welcome, Jagsy, to ETO!
Seems to me that a simple voltage regulator circuit, using the output of the convertor you listed, would suffice.
For instance:
View attachment 88624
In a SIM it only supplies the current that the RLoad dictates (the resistance of the LED strings) with about 1.3mA being used by the regulator.
A very simple circuit to construct and probably capable of handling most (if not all) of your strings (up to the limit of the convertor - 600mA). I'd attach the strings, adding one at a time, and see what happens. Note that the LM317 will get warmer as you add load and might need a heat sink.
And, given the usage, probably wouldn't need the 20uF cap. If a Vo of 1.25Vdc is inadequate, it can be raised by adjusting the circuit.