(Poisson/NegBin, RE) Estimation of time-invariant predictors in a panel model

Hello dear forum members,

For a time being I had been working with the following model (Stata command format):
- xtpoisson y x1 x2 x3 x4 x5, fe-

where, y(it) is a count DV, x1-x2(it) are controls, and x3-x5(it) are the predictors of interest, i -- denotes an individual (N = 1750), and t -- denotes year (2009-2015).
Note, I used POI-FE as suggested by Jeff Wooldridge, because of its robustness to over-dispersion and auto-correlation.

Now, there are new predictors of interest -- x6, x7, x8, x9 -- all of which were collected as time-invariant (in reality, x6/x9 do vary though, but it is not feasible to capture this variation). I did try some "panelization" strategies and use POI-FE estimator, however the consistency of the estimates was not satisfactory.

Therefore, I decided to keep the time-invariant predictors of interest (i.e., x6/x9) as is and estimate their effect using random-effects model specification:

-xtpoisson y x1 x2 x6 x7 x8 x9, re-
or (depending on the performance of the variance function)
-xtnbreg y x1 x2 x6 x7 x8 x9, re-

To ensure robustness of the estimates, I also consider GLM and QMLE (-ppml-) estimators.

Is this a plausible approach to estimate the effects of time-invariant predictors in a panel model?

A day spent exploring the solution leads to a night of (hopefully fruitful) reading. Looks like system GMM and/or Hausman-Taylor can do the trick with time-invariants better :)
I think of time invariants as fixed variables when you have longitudinal data. Is this what you are referencing or not?
Correct, Sir -- time invariants are fixed. The task in hand is to come up with an estimator that will (to a maximum extent possible) be consistent for estimating the coefficients of fixed variables.