My rant above is an attempt to argue that questions like this don't have answers, and what Andrew really wants is examples/counterexamples. So here is a counterexample, which may already be well-known to him. If $R$ is an $A$-algebra representating $\otimes_AM$ on $A$-algebras, and if $B\to C$ is an injective map of $A$-algebras, then $R(B)\to R(C)$ will be injective ($R(B)$ is the $A$-algebra homs from $R$ to $B$). But, for example, if $M=A/I$ then "usually" $B/IB\to C/IC$ is not injective (for example if $A$ is the integers, $I=(2)$, $B=A$, $C=A[1/2]$) so you're already dead in the water. My reading of the question is that it has been formulated to understand this sort of example so perhaps it's not a very good way of trying to understand it.

Here is an example where representability fails. If $R$ is an $A$-algebra representating $\otimes_AM$ on $A$-algebras, and if $B\to C$ is an injective map of $A$-algebras, then $R(B)\to R(C)$ will be injective ($R(B)$ is the $A$-algebra homs from $R$ to $B$). But, for example, if $M=A/I$ then "usually" $B/IB\to C/IC$ is not injective (for example if $A$ is the integers, $I=(2)$, $B=A$, $C=A[1/2]$) so you're already dead in the water.

Edit: emphasis of question changed, so ephasis of answer has been changed too.