Hausman test FE vs. RE in case that FE is not consistent

I have a twin panel data and want to estimate simple wage equation (interest in return to education). I use fixed effects(first differencing) to account for family background.(Based on: Ashenfelter and Kreuger 1994)

I have a strong economic argument that unobserved family invariant effect is correlated with the explanatory variables, implying that random effects are not a good idea. But I need to test it.

I want to perform the ordinary Hausman test. However, I know my education variable is measured with an error and thus is endogenous, causing FE estimator to be inconsistent.

Is the test valid in case FE is not consistent? Can I modify the test to work?

I have IVs (to correct the measurement error) at hand so my idea is to test FE_IV against RE_IV, but I am not sure if I can do it.

You are right that fixed effects is not consistent with measurement error and in fact measurement error is exacerbated in first differencing as is done in Ashenfelter and Krueger (1994). If a twin reports 12 years of schooling when she in truth has 13 years of education, the measurement error for random effects will be a difference of one year, i.e. less than 8%. For first differences that account for family fixed effects you subtract the schooling of twin 1 from the schooling of twin 2. Suppose twin 2 has 14 years of education, so with measurement error that's $14 - 12 = 2$ instead of $14 - 13 = 1$. This increased the measurement error already to 50% for the first differenced estimator.