Even his initial request for wisdom bothers me a bit. God, in a dream, offers Solomon anything, and Solomon asks for "an understanding heart / to judge your people [Israel] and to distinguish right from wrong." However, this is an echo of Eve's desire to eat from the tree in the middle of Eden as it "was desirable for gaining wisdom" and would allow her to "know good and evil." Solomon is supposedly rewarded for the same desire that brought sin to humanity (Yes, I know that sin was brought through defiance of God's law, but the intentions were the same).
Also, today's reading says that Solomon was given a "heart so wise and understanding that ... after you there will come no one to equal you." However, Jesus arrives later. Perhaps there is a bit of wordplay here which suggests that no one will "equal" him yet someone could "surpass" his wisdom, but I don't know enough of the original text and language to make that decision one way or the other. Still, he doesn't seem that wise.
The Bible gives us only one story depicting his wisdom, that of the two women who both claim a child as their own. The full story is tragic as both women had recently given birth, but one of the women had unknowingly smothered her child while she slept. She then exchanged her dead child with the other woman's living one (they were both prostitutes who lived in the same house, but otherwise alone).
Solomon orders the child cut in half, the real mother begs him not to and to give the baby to the other women so the baby can live, and Solomon thus awards the child to the woman who begged for the baby's life. Except, for this to work, the women had to truly believe he was going to kill the baby. Meanwhile, what if neither had said anything? Or if both had said that the baby should be spared? Or if the real mother would have preferred for the baby to die than to have him live with the other woman? The judgement showed an idealistic view of how people could react, but not necessarily a realistic one.
My biggest issue is that, despite his much vaunted wisdom, Solomon turns his back on God, even when God talks to him directly and warns him of the consequences of his actions. Certainly, Solomon creates the fabled temple and writes songs of praise, but his heart ends up being false, and all that he built falls to ruin. A truly wise man would have set up something that lasts, that allowed generations to follow in their praise of God.
Perhaps instead of seeking wisdom and understanding, Solomon should have asked for love and faithfulness.