It does not matter is the trait is continuos or discrete. If you see for example, the really nice and good book on Linear Models for the Prediction of Animal Breeding Values by Dr. R. A. Mrode, in chapter 3, page 44 ( I have 1996 edition). What, you can see is that fixed effects are being adjusted by random effects. Now, from the asumptions of the model, any random effect is assumed DNI ~( 0, Var(random effect)).
Perhaps, rather than adjusting the FE with RE why not try the 2-Stage FE (2SFE)estimation technique, this is quite similar to the Fixed Effects Vector Decomposition (FEVD). We all know that the FE is poor in estimation both strictly time-invariant variables (TIVs) and rarely-changing variables (RCVs) such that the estimator either drops them (TIVs) or make them insignificant (RCVs), thus reducing their explanatory powers. So you may have to run a 2-stage procedure. 1st using the FE estimator, regress your dependent variable on the time-varying variables and 'save' the residuals, 'u'. 2nd stage using OLS, regress the residuals on the TIVs and RCVs - the results you obtain will be close to using a RE estimator. I hope this helped.