Check for example Postel-Vinay and Robin (2002) and references therein to see how people go about structurally modeling the labour market.
The general framework is a dynamic one, where the job-search process is a Poisson arrival process for each worker, and the results are derived by finding the market equilibrium conditions.
The interesting point in this literature is that indeed labour market decisions are based on expectations about the employers, but here specifically the expectation about how career trajectories diverge and what this means for reservation wages for the employee, and how this depends on the productivity of firms.
In the cited paper, the productivity determines what a firm may be able to pay eventually. It raises wages when competing offers arrive by firms that may theoretically also employ the worker.
The worker forms expectations about his future trajectory and this implies that he is willing to work for a high productivity firm for less money, simply because he expects it can raise wages in the future if competing offers arrive.
So you see that expectations are indeed important for the job-search process. For your idea, you have to be specific as to what the worker expects if a firm is racist. An interesting idea would be to think about blacklists and stigma, which would mean that the worker could be locked-in the current firm. This would require the firm to offer higher wages compared to others.
It's a straightforward mechanism, and interesting I think, but you need to show basically how this impacts the workers choices based on preference (since you are doing structural modeling). It may be a good idea to start with reduced form analysis, and also to check out the sociology literature as I am sure this sort of idea has been used before. Your contribution would be to model it and get exact parameter values, run counterfactuals and hopefully discover some surprising results.
What worries me a bit about the idea is that you are somewhat missing an objective for the firm to be racist, given that it optimizes everything else in equilibrium in this setup. Of course you may assume it the same way one assumed productivity, and while it may be econometrically sound, it is not really identified in a theoretical sense. What I mean is that racism would just be a parameter akin to pushing down what a worker expects from a firm in terms of future productivity. And since it is outside of the explanatory power of a structural model, it may be that structural modeling is not really the necessary approach - a reduced form will tell you as much.
Perhaps you can make it bite, by making the stigma structurally different from lower productivity, and perhaps exactly this matters. This would be where the result would be interesting to me.
