Can we use hotelling's lemma as a rule of thumb when wage rate can vary?
I don't quite know what it means to apply a mathematical result to a situation in which the assumptions are violated, but the same idea as in the classical version of Hotelling's lemma can be used to understand the marginal effect of wage changes and you can combine the two to understand what happens on the margin if you change both prices and wages.

The underlying logic of Hotelling's lemma was that for a profit-maximizing firm, the marginal effect of adjustment outputs (and therefore inputs) is zero. This tells us that the marginal effect of wage changes on profits should be proportional to inputs used, that is
$$\frac{\partial \pi(p,w)}{\partial w}=-x(p,w),$$
with $x(p,w)$ the optimal input bundle. This is indeed true and the proof is exactly the same as for the marginal effect of $p$, just replace the derivative with respect to $p$ by the derivative with respect to $w$. If $f$ is the production function, the profit function is given by
$$\pi(p,w)=pf\big(x(p,w)\big)-wx(p,w).$$ 
Taking the derivative with respect to $w$ now gives us
$$\frac{\partial \pi(p,w)}{\partial w}=pf'\big(x(p,w)\big)\frac{\partial x(p,w)}{\partial w}- x(p,w)-w\frac{\partial x(p,w)}{\partial w}.$$
The first order condition for $x(p,w)$ being a maximizer of $pf(x)-wx$ implies
$$pf'\big(x(p,w)\big)- x(p,w)-w=0$$ and therefore
$$pf'\big(x(p,w)\big)\frac{\partial x(p,w)}{\partial w}-w\frac{\partial x(p,w)}{\partial w}=0.$$
Substituting this in, we get
$$\frac{\partial \pi(p,w)}{\partial w}=-x(p,w).$$

The envelope theorem generalizes the underlying idea that marginal benefits of adjustments for optimizing agents are zero for more general parameterized optimization problems.