To answer the question of "can you hear it". Dunno, no one can hear through your ears.
Consider these factors: A well made, no sonic contribution box, is well integrated and damped. The easiest way to get there is MDF + No Rez. Theoretically, voids in plywood would act as small resonant chambers. Most hardwood veneer plywood has softwood core plys that are certainly less dense than hardwood or MDF. Baltic Birch has hardwood ply cores that are same as finish plys.
Where does that leave us in the real world? Generally, substitution of design materials has an effect. Is it audible to you? Dunno. I usually think in terms of "details matter", and that leads me to follow or improve upon known design criteria. Leaves nothing to chance, ya know? But that's my way, doesn't mean it suits everyone.
I have zero knowledge of Big Box policies concerning cutting or concerns about how to transport so all the above should be taken with appropriate grain of salt for your situation.
In the end, words, words, words
