Maybe throw a brace on the inside of each side between the front and back and another 2 evenly spaced through the center, level the top out then I'm sure she'll carry more than enough, I prop roofs up with less than that
These posts were 120mm x 120mm (or 5x5 as you muricans say) holding up a lot more than 700lbs when completed with people on the balcony plus the weight of the deck and roof, just need to get the top 100% flush..
Ps. I'm a carpenter
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