I need to get back into the office (where I have some of the circuit's schematic - there are bits missing) and I'll try and post it up.
Most likely, it is less than 40mA but there is an LED connected in there somewhere so I'd rather play it a little safe.
I'm beginning to think that a little 1A 5v relay might be a pretty easy way to do it after all. The 4ms delay wouldn't matter at all in this case and it'd give me a simple way of doing exactly what the switch does, but controlled from the duino.
Certainly learning a lot from this little project... and from all your help.