Hi Wend - apologies, it seems I got the wrong end of the stick re the heads touching. I got the impression that the trainer had told you to put the bowls down right next to each other, which now seems not to be the case. It would only be in that instance that I think the lead thing could escalate the problem if they started to fight - as you describe it, the trailing lead sounds perfect for the job.
Re Claires bowls, it probably depends on what type of flooring you have - I use the same bowls but my kitchen floor is rubber/vinyl and they don't shift at all (unlike the plastic PAH ones I had which needed a bit of carpet under them). Maybe if you got a couple of rubber car mats, or something similar, it might help.