I honestly can't think of a good argument against it.
You should really think harder then, but barring that you should look up ANY example of polyamory online. It's always associated with misery, pain, and overweight gender blobs.
Polyamory just doesn't work.
so there's not even a strong religious case against it.
Abraham caused a generational blood feud because he let his wife convince him to impregnate her servant and his wife HATED her servant for it so much after the fact that she demanded Abraham cast her and her son out into the wilderness.
Jacob married Leah (by accident and deception on the part of Leah and her father) and then her sister Rachel (the one he actually wanted and was willing to work FOURTEEN YEARS to earn her father's permission) AND their two maids because the sisters were so jealous of each other they were constantly fighting over who was the better wife. The family conflict directly resulted in Joseph (coat of many colors fame) being nearly killed by his half brothers and sold into slavery.
Instead of getting to settle into building the temple, King David's reign was defined by conflict, rebellion, and retarded family drama including his daughter being raped by her half brother because he took multiple wives.
Which set the example for Solomon, the wisest man alive to that point, who lost the favor of God because he collected a harem of women who all worshiped foreign gods and led him astray.
The Old Testament is one example after another of God's chosen people thinking polyamory will totally work this time only for it to blow up in their faces.