You will want to be careful with the priors. A key feature of the econometric fixed effects model is that the respondent-specific intercept is allowed to covary freely with all time-invariant variables. This is what makes the fixed effects model so powerful, because those intercepts stand in for all time-invariant variables (measured or unmeasured) with time-constant effects. Per Paul Allison (Fixed Effects Regression Models, 2009, page 21-22), a random effects model treats the random intercepts as uncorrelated with time-invariant variables, which is not what you want.
The only sure approach I can think of is to place wide, independent, uniform priors on the respondent-level intercepts, at least to align with the equation you cite and the traditional approach of including a dummy variable for each respondent. But there is probably some greater nuance here that might lead to more informed priors being appropriate.