Ah I see - it looks like LsqFit is doing an explicit promotion to float somewhere. Not sure whether one should expect LsqFit to work with Measurements - maybe @pkofod can comment.
What do you expect to happen with the uncertainty in estimation? It looks like measurement error in the covariates (right hand side) of your equation to me, which traditionally in econometrics would just lead to attenuation bias. Or do you have different (known) degrees of uncertainties in different parts of your data set? In that case this discussion here might be of relevance, which shows the use of inverse variance weights in LsqFit: