Do you intend system $A$ to include only $I\Delta_0$, or actually $I\Delta_0(E)$? That is, do you allow induction for formulas involving $E$?

If you do allow it, then then you can prove easily $E(x,y)\to\exists z\le y\\,z=2^x$, hence the theory proves exponentiation to be total.

If you only allow induction for $\Delta_0$ formulas without $E$, then the answer is negative, and in fact, $A$ is a conservative extension of $I\Delta_0$. This can be seen as follows: take any model $M\models I\Delta_0$, we’ll expand it to a model of $A$. For $n\in\mathbb N$, put $E(n,2^n)$. For nonstandard $x,y\in M$, let $x\sim y$ iff $x-y\in\mathbb Z$. For any equivalence class $C$ of $\sim$, choose its representative $a$ so that $C=a+\mathbb Z$, and choose an arbitrary nonstandard $b\in M$. Then for any $n\in\mathbb N$, put $E(a+n,b2^n)$ and $E(a-n,\lfloor b/2^n\rfloor)$.