What’s going wrong in the former case? Can it be fixed without eliminating the \mu_t^x (because I want to expand the model in such a way that eliminating the Lagrange multipler by hand may be difficult).

Using the third equation - i.e. E_t \mu_{t+1}^x = 0 - I get \frac{1}{\beta} \mu_t^x = E_t x_{t+1}. Substituting this expression in for \mu_t^x and \mu_{t-1}^x in the second equation then gives

E_t x_{t+1} = \frac{1}{\beta} (E_{t-1} x_t - x_t)

which is the second equation in the second system.

Conversely, to go from the second system to the first system, I define \mu_t^x = \beta E_t x_{t+1}. Substituting into the second equation of the second system for E_t x_{t+1} and E_{t-1} x_t gives the second equation of the first system. Further

You can also stay with Dynare 4.6.4 for the moment.
You can prevent the preprocessor bug by tagging the third equation for \mu_x.
E.g. this model block should solve the problem