Is it cardinal to have background in math/physics/engineering in order to be able to work professionally in acoustics?
It depends, because I've heard some sound technicians working on studio acoustics with good results. But I'm quite positive that even they've done some reading to do it properly.
However, for something more complex than a small studio space (imagine designing a concert hall or a museum space for example, or anything for which there are no "simple formulas"), I'd say that it's mandatory to understand sound's physical principles in order to acoustically design and treat spaces. You would be involved in the building phase already, so it's not just about "acoustically treating existing spaces", but it could be about building spaces for the correct acoustics. It's not guesswork, but deeply rooted in known physical phenomena (waves, signals, mechanics, structures, materials) and possibly also to following some regulations (e.g. sound levels, permitted materials, structural regulations).
Edit: also, if you're thinking about doing contract work, then not having an appropriate educational background (e.g. acoustics or structural engineering, math or physics) might severely limit your scope of work or it might be even externally restricted ("professional engineering" is regulated in some countries).