Yes (if I understand the question correctly), you can use posterior_predict or posterior_linpred (the former includes the observation nose, the latter only the model uncertainty). You can also predict for new data. One method I’ve found useful in some cases is to make predictions for each person while varying only the factor of interest - see PPC for ordered_logistic? - #2 by martinmodrak for a bit more detail and ideas by others.
In Bayesian models it is usually beneficial to work with all the posterior draws instead of just the mean estimate, although this can make the post processing a bit more involved. predict and fitted methods of brmsfit also offer setting summary = TRUE to get a single estimate if that’s what you prefer.
I believe @torkar might have interpreted your query as how to model individual-level variation, instead of obtaining individual-level prediction which might have led to some confusion (unless I am the one interpreting the conversation wrong :-) )
The difference in the model coefficients themselves is slightly surprising, but hard to interpret on its own as the intercepts have also changed noticeably. As you have correctly noted, using prediction is usually better for comparing differences.
Best of luck with your model!