I am trying to find out if I should use a Fixed Effect model versus a Random Effect model. Given a scenario.

To summarize, I am trying to measure profit as a dependent, with the 1 being the item return policy across several different stores of the same organization. There are different return policies across the different stores, some more lenient than others. There are other explanatory variables, and there are latent constructs - which necessitates the need for either Random Effect or Fixed.

My question is, if I specifically state that "the other elements of the return policy remain constant across all stores" Other elements meaning excluding the leniency of the return policy.

Because Fixed Effect is unable to have time constant explanatory variables, does the above statement disqualify the Fixed Effect model?