Are Parameter Estimates Unbiased If the Dependent Variable is a Per Capita Rate Based on Approximate Population Data?
What about assuming a multiplicative error process and then using logs? 

Say the data generating process were a little different:
$Y_i  =  \beta_0 \cdot  X_i^{\beta_1}\cdot E_i$

If $Y_i  =  Z_i / P_i$ in truth but all we could really observe was:

$\hat{P}_i = P_i \cdot \Gamma_i$, where $P_i$ was the true value and $\Gamma_i$ was a strictly positive measurement error.

We' be estimating the following equation in practice:

$\hat{Y}_i  = Z_i / \hat{P}_i =  \beta_0 \cdot  X_i^{\beta_1}\cdot E_i$

take the logs of both sides (lower case letters are logs of individual variables):

$y_i  = \beta_0 + \beta_1 \cdot x_i + \epsilon_i - \gamma_i$

If we define a variable $\xi_i = \epsilon_i - \gamma_i$ then we can have an equation $y_i  = \beta_0 + \beta_1 \cdot x_i + \xi_i$ that looks very much like the one you wrote above. 

If $\xi_i$ satisfies the same relationships with the log variables that $\epsilon_i$ does with the level variables it seems like everything should still work.