For me, it usually takes no more than 2 cubes to do it.
I hold the cube in my hand until it melts, then grab another cube.
Given how small the cube is, I've never made them with RODI water in advance. I don't think a couple of cubes would do any harm.
If you are worried about ice cubes melting, I guess you could also do it with a mini gel pak; but, if it leaks that would be much worse.
Best of luck,
Roy