I don't know the answer to this question, but
will make an extended remark.

Let $X$ be a finite set and let $f:X^n\to X$ be any $n$-ary operation
on $X$, $n>0$.

**Claim.** The following conditions are equivalent.

(i) $f$ is surjective with uniform kernel.

(Equivalently, for each $a\in X$ the set
$f^{-1}(a)=\{(x_{1},\dots,x_{n}):f(x_{1},\dots,x_{n})=a\}$
has size $|X|^{n-1}.$)

(ii) There exist $n$-ary operations on $X$, $T_2,\ldots, T_n$
and $S_1,\ldots, S_n$ such that, if $G,H$ are
$$G(\bar{x}) = (f(\bar{x}), T_2(\bar{x}), \ldots, T_n(\bar{x}))$$
and
$$H(\bar{x}) = (S_1(\bar{x}), S_2(\bar{x}), \ldots, S_n(\bar{x})),$$
then $G$ and $H$ are inverse bijections between
$X^n$ and $X^n$.

The question asks, if $V$ is a variety satisfying:

I. $V$ is finitely axiomatizable.

II. $V$ is generated by its finite members.

III. Item (i) above holds for the interpretation
of any fundamental operation of arity at least $1$
on each finite member of $V$,

then must Item (ii) above hold in the strong sense
that the $S$'s and $T$'s are term operations, but in the weak
sense that we allow other parameters $m$ and $r$ in place
of some instances of $n$?

Roughly, this asks if having Item (i) hold throughout
the finite part of $V$ implies that Item (ii) is enforced
by the equational theory of $V$.

This seems plausible to me, but it also seems that there are some extraneous
elements in the question. I don't think that $V$ being
finitely axiomatizable is relevant. I don't think the additional
flexibility of introducing parameters $m$ and $r$ possibly different from $n$ helps, but I haven't tried to check any examples.
(It is clear at least that $m$ must equal $r$ if $V$ is generated by
its finite members.)
I also think the result, if true, is not a property
of varieties; that is, the question can be asked for a single
(fundamental) operation of $V$: if $V$ is generated by its finite
members and $f$ is a fundamental operation of positive arity
satisfying Item (i) above, then must Item (ii) above hold?

Here is a sketch of a proof of the claim.

(ii) implies (i):
Let $\pi_1: X^n\to X$ be the first projection map.
It is surjective with uniform kernel. Since $G: X^n\to X^n$
is a bijection, $\pi_1\circ G$ ( = $f$) is also surjective with
uniform kernel.

(i) implies (ii): For each $a\in X$, choose a bijection
$\beta_a: f^{-1}(a)\to X^{n-1}$. (Item (i) is
the statement that such a bijection exists.) If
$$G: X^n\to X^n: \bar{x}\mapsto (f(\bar{x}),\beta_{f(\bar{x})}(\bar{x})),$$
then $G$ is a bijection, and the appropriate component functions
exist for both $G$ and its inverse $H$.