A day surface action in the Pacific was a very rare bird. I think the possibility is extremely remote to be honest. The calculations for an intercept dont take into account course changes or the current fuel status of the US fleet. I understand some elements had started the operation from Pearl, some from Kwajalein, so chances are that at least a portion of the US fleet would be short of fuel.
A rough comparison might be to compare the RN experiences in hunting down the Bismarck. Only after the Bismarck had been immobilized by carrier strikes was it possible for the British to close for the surface engaement. And even though the Bismarck was easily within range of the RN bases in terms of the ships endurances, the need for constant changes of course and speed, meant that by the time the interception had been effected, the British ships were running dangerously low on fuel. Bottom line is that it is extremely difficult to engage an enemy with surface forces that does not wish to be engaged
The difficulty also with this scenario is that at least a portion of the USN would need to be kept back to protect the amphibious forces.
So even though the USN outnumbered the Japanese, it is highly unlikley that in an actual battle that the full strength of the USN could be used. The DDs in particular are highly unlikely to make it.
In a stern chase situation, one might expect the Iowas to push ahead of the main fleeet, to attempt to slow down the main portions of the Japanese fleet. Whilst their top speeds were 33 knots, their best sea speeds were no more than 30 knots (commonly US reported speeds are the speeds as new, not taking into account refit, engine wear and the like). Against this the sea speed of the Mobile Fleet was 24-26 knots. The Iowas would therefore be closing at just 4 knots, or 8000 yds per hour. Assuming the Yamato and Musashi were detailed as Stern Guards, they would commence firing at a whopping 42000 yds. The immune zone of the Iowas against 18 shells ceases below about 36000 yards (outside of that, the belt would not be penetrated, but they still risk deck penetrations, or damage from fire or "special damage"....like the Hood). In exchange, the 16 in shells cannot really hurt the Yamatos until the range is something like 20000 yds (more or less). That means that the Iowas must endure a sustained bombardment for more than 2 hours. If they heave to for a full broadsides, the time is even longer, as a general rule, if this broadside to approach is adopted, the rate of closure is about half that of a direct, Nelsonian charge. So the Iowas are going to be under attack for something like 3 hours before they can effectively return fire.
The South Dakotas have an effective sea speed of only 27 knots, so they are not going to reach the battle. The destroyers will have to reduce speed to conserve fuel. The cruisers can keep up with the Iowas, but they will need to close to under 15000 yards to be effective, and that exposes them to concentrated Long Lance attack.
This appraisal is far from complete, but it should be obvious that when you look at the detail, a surface engagement is most unlikley to favour the US, in the circaumstances presented at Phillipinnes Sea.