Consider the following model: $$y_t = c_0 + \phi y_{t-1} + \beta x_{t} + u_t,$$ where $u_t$ follows an MA(1) process: $u_t = \varepsilon_t + \theta\varepsilon_{t-1},$ where $\varepsilon_t$ is white noise with mean 0 and variance $\sigma^2$. The variable $x_t$ is exogenous in the sense that it is uncorrelated with $u_t$. This is essentially an ARMAX(1,1) model that has been misspecified as an ARX(1) model. I want to fit this model using OLS. Now say that I don't really care about estimates of the intercept $c_0$ and the autoregressive parameter $\phi$, but I am concerned about the asymptotic properties of the estimate of $\beta$. I know that as I have autocorrelation and a lagged dependent variable, OLS is inconsistent. Is there an expression in the literature for the asymptotic bias of the OLS estimator of $\beta$ in this case? Ideally without assuming normality of $\varepsilon_t$. Further, would applying feasible GLS with a consistent estimate of $\theta$ aid the situation in any way?
