Actually I think it would be unwise to put that condition on a mixer.
It reminds me of the method eventually used to break the enigma machines. The breakers worked out that the enigma machine would never encode a letter to itself. That gave them enough of a statistical hole that they could begin breaking.
In the same way, one could simply look at which output coins a person received that contained none of a particular input and voila, the connection is made from input to output.
The correct solution is that coins are completely randomly mixed.
You seem to describe a service where I would deposit coins, check to see that none of the original came back. If they did, I deposit them again later after the service has recharged a little, check again. Meanwhile I'm paying commissions each time. This would be a nuisance.