This is more "thinking out loud" than any kind of firm guidance, but it might perhaps be of some help.
let $y=\text{price}$, and $x=\text{quantity}$. Then your model says something like
$y \cdot x = \beta_0+x^\beta_1 +\eta$
but nothing was stated about $\eta$. I assume that for a multiplicative-type variable likes sales the variance would be roughly proportional to the mean (and your mention of logs encourages me in that line of thinking). In that case:
Dividing through by $x$
$y = \beta_0/x+x^{\beta_1-1}+ \eta/x = \beta_0 x_0+ x^{\beta_1^*} + \epsilon$
Now if on this scale you can regard the variance as nearly constant, you could fit the equation via nonlinear least squares. Alternatively you could do some form of variance modelling, re-weighting via IRLS, or variance-adjustment if necessary; I'm not sure what would be seen as suitable for what you're doing (but it may not be needed if the variance specification it broadly reasonable).
now $\frac{\partial y}{\partial x} = -\beta_0/x^2+ (\beta_1-1) x^{\beta_1-2}$
and so the reciprocal of own price elasticity would be
$\frac{x}{y} (-\beta_0/x^2+ (\beta_1-1) x^{\beta_1-2})$
$=\frac{1}{y} (-\beta_0/x+ (\beta_1-1) x^{\beta_1-1})$
and hence I think the elasticity you want would be
$\frac{y}{-\beta_0/x+ (\beta_1-1) x^{\beta_1-1}}$
or
$\frac{y\times x}{-\beta_0 + (\beta_1-1) x^{\beta_1}}$
... which you can evaluate at the mean or whatever as needed.
