"Should"?
As in legally required? Heck no!
As in to remedy health, posture, and comfort issues? sure, if it helps it helps!
To please fantasizing kinksters' aesthetic or erotic preferences? Absolutely not.
To please their own sense of aesthetics, or to feel the sensation? Sure!
You know the cross strap found on some backpacks? I use it mostly because it gives a sensation close to wearing a simple harness

I know a bra can do a similar thing - just a matter of fonding the right shape and materials.
Bow I am not an expert on wearing bras, but I know a little bit of physics.
Thing is, assuming mermaids spend a majority of time submersed in water and more or less neutrally bouyant, I guess similar effects hold as they do in weightlessness, and by the equivalence principle, feeling weight of boobs only happens during acceleration. There is also drag, roughly square in velocity that presents as a force akin to weight. Finally inertial delay from loose coupling of breasts with the skeleton (jiggling through turning). A bra could remedy that given the right materials - needs to be stiff (rigid) enough to transfer force but elastic and soft enough to not cause discomfort.
In conclusion I think wearing bras underwater is not a "should" but more of a suggestion if more rigorous movement is conducted.
I would love to read what (sizable) boob-havers have to say on the subject of going bra-less underwater.
If I was a mermaid with boobs.. hm, maybe I'd wear them to conductthese experiments, or maybe for collecting delicious sea food while I swim. Perhaps for the aesthetics or sensation.