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

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

**Claim.** The following conditions are equivalent.<p>

(i) $f$ is surjective with uniform kernel. <br>
(Equivalently, for each $a\in X$ the set 
$X_a=\{(x_{1},\dots,x_{n}):f(x_{1},\dots,x_{n})=a\}$
has size $|X|^{n-1}.$)<p>

(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 $X^n$.<p>

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

I. $V$ is finitely axiomatizable. <br>
II. $V$ is generated by its finite members. <br>
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 whether if having Item (i) hold throughout
the finite part of $V$ implies that Item (ii) is enforced
by the equational theory of $V$.<p>

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 $r$ helps
(but I haven't tried to check any examples).
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?