I'm with 2nd WI and Mr. Steele is quite correct: for us, the dress hat from Tim Bender is the way to go! They block the sun nicely, can be reshaped as one might wish, and his repros are about as accurate as you can get. Of 54 Wisconsin regiments, only 11 served in the East with the Army of the Potomac, with the three Iron Brigade regiments donning the Black Hats; I have also seen good representation of those hats in the 5th. The rest were serving in the West, making the dress hat the far more likely.
However, I will recommend that you eventually purchase a forage cap as well (instead of the kepi), especially if you'd like to join the Black Hat Battalion. Depending on scenario, the men might be asked to portray a different regiment for the purposes of a scenario, in which case doffing the tall felt dress hats in favor of standard AotP infantry caps is necessary. The reason I recommend the forage cap is that, despite some regiments adopting the kepi for all ranks, most were issued with forage caps and that is what they tended to maintain.
My thoughts-- feel free to comment, gentlemen!
Most Respectfully Submitted,
-IronBrigade'63