The BBs would have been maneuvering so it all comes down to naval gunnery, who has the most accurate guns. This goes to Iowa class hands down. Take for example on March 20, 1942 In the Inland Sea, Admiral Yamamato conducted armament trails and they were judged a failure. Both captain Takayanagi and his gunnery officer were called fools because the Yamatos gun aimers manning the rangefinder misread the horizontal settings. Now compare this to the Iowa who had the first ever TDC and could easily hit targets at 24 miles. The Iowas MVII AP rounds could easily penetrate the Kongos belt armor b/c it was faulty. The Yamatos belt armor proved to be a failure. On December 25, 1943 one torpedeo hit showed a total failure of the main armor belt due to a flaw in the lower side protection belts. She took on 3,000 tons of water and the 3rd magazine was flooded.
Speaking to the accuarcy of Iowas guns one must remember her fire control radar. The radar was amazing. Officers testing the equipment aboard USS Iowa wrote:
"Spotting both 5-inch and 16-inch splashes, HC or AP, with the radar is comparable to deliberately drawing a picture of the splashes on paper and looking at it. At all ranges fired during this period, the most inexperienced officer, given a brief explanation of what to expect, can spot splashes accurately to within 100 yards, and to within 50 yards with some experience."
The Kongo class BBS had no radar range finders and did not have the complex TDC that the Iowas had. The Yamato in a night engagement was toast, even fighting in broad daylight it is obvious from the example that her gunners were incompetent. Yamamato couldn't determine range and keep a plot of a moving target to save her life. Iowa was automatically updated by the best TDC computer and radar on any BB ever put to sea. Yamatos armor was proven faulty when hit in action showing that the Iowa would have made short work of her in day or night gun duels.
Yamato vs. Iowa
Staring with the obvious, Iowa was faster (27 kts. vs. 33+ kts.), and by battleship standards, quite a bit more limber. The speed advantage here gives Iowa a very important tactical advantage in which she easily chooses the range of the engagement. An important consideration when she (as Yamato) can shoot over the horizon.
Which, of course brings me to fire control, Iowa (as every other US Navy Ship larger then a Destroyer at the time) used the Mk. 13 Fire Control suite. The system was insanely ahead of it’s time it had a 3cm wavelength to Yamatos 10cm, and almost 25 times the power output. For Iowa this means it not only can shoot over the horizon, but can find a perfect firing solution over the horizon, and adjust it with pin point accuracy (by Battleship standards), while maneuvering. To read about this system in action read up on the 3rd Battle of Savo Island. Yamato had the same range-finding set up as Kirishima (Bracketing a target with colored smoke to saddle a slavo... etc)... And Iowas was certainly superior to the set that was fitted to the Washington. Having the best range finding optics was about as useful here as the Mk I eyeball used at Jutland.
As far as Iowas 16"/50 main battery, I consider it to be the equal to Yamatos in all practical terms mostly because of the projectiles it fired. It would have been able to pierce Yamtos hull eventually and wreak havoc on her decks right from the start. The plunging fire on her decks would have been a real concern, and even though Yamato had the thickest armored decks around, you would simply be asking to much of that armor when Iowa was dropping her shells in at 20,000+ yds with deadly accuracy.
Iowa was also no slouch in the Armor dept. and even though she had an “all-or-nothing” approach, her actual armor layout was superb with all of her important machinery and armament well protected. This does not even begin to discuss the qualative difference in steels that both ships were using, US steel being far less brittle then the Japanese steel of WWII. Meaning even if Yamato got very lucky, and hit a ship that it could not see, Iowa would certainly survive the hit and continue lobbing shells at Yamato.
Simply put, Iowa had a far better effective range, and she chooses the range of the fight. She wins in almost any conceivable scenario.
As stated earlier, Bismark was built for Jutland style knife fights of the North Atlantic, he could make it a tough fight for almost anyone if he could close the gap. But, he also had a fatal design flaw where her primary hydraulics sat above significant armor protection, tisk, tisk.... I don’t think that realistically he would fare well against either, as he would have to get in close to effectively engage either one; a situation Iowa would never give her, and Yamato would welcome.
A more interesting fight, in my mind, would have to be KM Bismark vs. Richelieu as they were on each others minds when they were being developed, and both were launched in ‘39. I think these two were natural advisaries myself.
And of course the pound for pound armor king H.M.S. Prince of Whales vs.the fast, deadly and ever so useful IJN Kongo.
If I had a trillion bucks I would build both ships, automate them, give the remote control to some naval experts and sit back and watch the death match.