If by "the usual axiomatization" you mean the implications,
$x/y \leq z \Leftrightarrow x \leq zy \quad$ and $\quad x\backslash y \leq z \Leftrightarrow y\leq xz,$
and if you mean you want an axiomatization that does not involve implications, then I believe the following theorem, due to Blount, Jipsen, and Tsinakis, answers your question:
Theorem. An algebra is a residuated lattice if and only if it satisfies the equations defining lattices, the equations defining monoids, and the following four equations:
- $x(x\backslash z\wedge y) \leq z$,
- $(y\wedge z/x)x \leq z$,
- $y \leq x\backslash (xy\vee z)$,
- $y \leq (z\vee yx)/x$.
Thus the class of residuated lattices does form an equational class. (If you don't like the appearances of $\leq$, then, as you say, you can replace them using an extra meet operation.)
This version of the theorem appears in Galatos, Jipsen, Kowalski, Ono MR2531579, page 94.