It seems, perhaps obviously, that "construe" and "construct" have nearly identical etymologies. Since that is true, is there a reason--as for "use" instead of "utilize"--that one should use the more minimal "construct" (instead of "construe") at all times?
Construe is for interpreting meaning or, much less often, to combine words into a grammatical sequence.
Construct has a much wider range of applicable settings, and is centred around the meaning of building. (And indeed, the principal use is in the building of things like buildings).
Generally speaking, we would construct a sentence, and construe the meaning of the already-constructed sentence.
I have never seen construe used in the sense of to construct [a phrase] out in the wild. So the principal reason to use "construct" is that people will fail to construe your meaning, if you use "construe".