Can the isomorphism relation for countable models become harder when adding finitely many constants?

Can the isomorphism relation for countable models become harder when adding finitely many constants? References I am particularly interesting in the case where $T$ is o-minimal, but I would be interested in any theory $T$ (or even an $L_{\\omega_1,\\omega}$-sentence) which has this property. Context: view the set of countable models of $T$ (under isomorphism) as an equivalence relation on a Polish space. Take a finite set of parameters $A$ (from some model of $T$) and then consider the set of countable models of $T_A$ (under $A$-isomorphism) as another equivalence relation on another Polish space. One could reasonably ask if the first is Borel-reducible to the second, or vice-versa. In every example I know of, both are Borel-reducible to each other. However proving this in general has eluded me. I believe that in the general case (or even in the first-order-theory case, or the o-minimal case) this is an open question. So I would like to know if this is the case, and either way, a link to any published results on this topic (or relevant examples either way). Thank you!