Dec. 4th, Grant to Halleck --
How far south would you like me to go? ... with my present force it
would not be safe to go beyond Grenada ...
Dec. 5th, Halleck to Grant --
... you should not attempt to hold the country south of the
Tallahatchie ... your main object should be to hold the line from
Memphis to Corinth with as small a force as possible, while the largest
number possible is thrown upon Vicksburg with the gunboats.
Thus ends Grant's Overland Campaign for Vicksburg, sabotaged by a conspiracy of dunces: Lincoln, Halleck, and McClernand. Per Halleck's instructions, Grant detached the largest number possible back to Memphis to be thrown at Vicksburg via a riverine operation. The remainder of Grant's force would hold Pemberton at Grenada and pursue if Pemberton fled south.
Sherman's message is based on the assumption that if a heavy force arrived on the doorstep of Vicksburg, Pemberton would do just that thing: abandon Grenada and flee south. This assumption is a counter-factual: Pemberton did not abandon Grenada. In fact, before Pemberton knew beyond doubt that Grant was moving back to Memphis, he detached a mere two brigades back to Vicksburg. Vaughn's Brigade was detached immediately, Gregg's a couple of days later. These two brigades would be the only reinforcements to be involved in Chickasaw Bayou. Vaughn's Brigade experienced some light skirmishing south of the Indian Mound position and Gregg's Brigade sat in reserve. Sherman's message is a bit laughable in this regard. Sherman says that he observed "the heavy reenforcements pouring into Vicksburg." While Gregg's Brigade was certainly a force to be reckoned with, Vaughn's Brigade was a smallish brigade that was heavily populated with conscripted East Tennesseans. The simple fact is that Sherman with four divisions was bested by Stephen D. Lee with a single ad-hoc division, and arriving reinforcements contributed little to this end.
So Sherman is ultimately repulsed at Chickasaw Bayou, and Pemberton remains at Grenada with his principal force. Therefore I can see no counter-factual that has Grant moving overland past Grenada. Halleck was telling Grant to withdraw to Memphis and Corinth. Grant's right-hand man had been advising him to move back to Memphis and take command of the principal force. It would appear that news of Sherman's failure and the arrival of McClernand would have resulted in exactly the movement that Van Dorn's Raid prompted ... so the great irony of Van Dorn's Raid is that it expedited Grant's eventual decision to move with McPherson to Milliken's Bend, and therefore ultimately expedited the fall of Vicksburg.