In the context of my hypothetical dataset, what are the potential sources of measurement error that might affect the validity of my structural equation model, and how can I address these issues?
There are two types of measure error: random (which is usually treated as a problem of reliability) and systematic (which is usually treated as a problem of validity). If your SEM includes unmeasured variables (i.e., a measurement model), then this will take care of problems with reliability. The primary method for assessing validity is through contract validity, which examines predicted versus observed patterns of correlation.
the error in a latent variable model consists of both random and systematic error. It is nothing different from any other error term (of any structural equation or causal effect). And as in any other system, common (systematic) errors, that is confounding, is a problem.