All of these 2+1D dualities are IR dualities, which hold after turning on all relevant operators which respect the symmetries and for lack of a better word, "phase constraints" like vanishing expectation value of some relevant charged operators like $\phi, \phi'$, which are insensitive to small enough perturbations. So you can imagine $V$ contains terms like $m^2 |\phi^2| + m'^2|\phi'|^2$ and the condition on the expectation value is saying $m^2 , m'^2 > 0$. $m^2 = 0$ for instance would be fine tuned, and one would need to locate in the generically dual theory the relevant operator corresponding to $|\phi^2|$ and tune it also. Sometimes this works and I don't think anyone knows a good argument.
Then, if we just want to see what the spectrum of the theory is like in the IR, we can replace these massive fields with their expectation values (which is just zero for both). What remains is a Chern-Simons-Maxwell theory which is known to be gapped. See this paper about "topologically massive gauge theory".