*(I'll let someone else address the estimation of the missing data.  You may want to model the probability that the observation is each level of the unknown factor directly using knowledge of other covariate values, and possibly outside information, e.g., priors etc.  However, at first glance your approach looks reasonable to me.)*  

One note is that I can't tell from your description if you are weighting by *raw frequencies*.  If so, you want to divide these by $N$ to get the marginal *probabilities* instead.  

You are right that you are not handling level 3 correctly.  The coding scheme that you use in your question set up is known as *reference level coding*.  To use this approach correctly, you need to have an intercept (i.e., $\beta_0$), which estimates the mean of level 3.  I suspect you do have such, even though you didn't list it.  In this case, you would just add the intercept to your final equation.  That is: 
$$
\beta_0\!*\!f_3 + \beta_1\!*\!f_1 + \beta_2\!*\!f_2
$$
Note that you are multiplying the intercept (which encodes the reference level) by the marginal probability that the observation is actually the reference level.