So instead of just theorising... I went and tore this thing down. It's a simple, yet clever design that produces multiple different voltages by using batteries at different points of the bank.
If I had just rigged up a 15V power supply, nothing would have worked.
I discovered there was more going on while putting some batteries in - the fan and LEDS started up before I was done putting in batteries, since I'd bumped the on switch at some point, unintentionally.
Within the battery compartment, there are actually two separate circuits. One pair of batteries is purely for the motor that rotates the multiple bubble rings. These rotate and fall into the bubble mixture as they pass a holding vat, and then continue on around the merry go round over the main fan.
There's a 6V circuit that draws partially from the lower 4 batteries and drives a strip of 4 multicoloured LEDs.
The main fan uses a 12V supply, using all 8 of the other batteries. It indeed draws >500mA, although I can't measure beyond that at the moment while I wait for my replacement multimeter fuses to arrive. (The 10A circuit on my MM is presently fuse-less, owing to an act of stupidity several years ago).
I've sketched up the actual circuit now:

So now my plan is to use a couple of DC-DC converters and a 12V 2A supply, which realistically should have enough oomph.