If Meade sent this to Halleck on the 3rd, evidently he had it on the 2nd.

There would have been at least two reasons, in my thinking, to send these to Halleck. First, as an informational base to inform Halleck. Second, as a warning to Halleck to look to Washingtons "backdoor", so to speak, and not to become fixated upon just what was going on at Gettysburg. So Halleck could assemble a force around Washington should such a threat develope.

Do you think that these dispatches were a part of the reason that Meade failed to pursue Lee's retreat from Pennsylvania for 9 days?

Certainly the Union Army was not in any real condition to follow Lee's and to fight anthr decisive battle against Lee. Especially with Lee's army on the defense and Lee picking the defensive position. I know that is the popular opinion of most historians. But, could these dispatches have had the effect of "freezing" Meade in place, while Lee retreated, on the possibility that Lee might recieve these reenforcements anyway and turn on Washington, If Meade was out of position?

I realize that these dispatches, almost too clearly, indicated that this force was not going to be sent to Culpepper. BUT, Meade and Halleck would have also looked at this as being a ploy, a "Bait and Switch", or misdirection tactic, so to speak. It wouldn't be the first time in history that misinformation would have been given to an enemy to confuse them.

Do you think it may have immobilized their thinking into a "wait to see" attitude, to see if another threat would materialize if Meade pursued too quickly?

I wonder what would have happened, IF such a misdirection dispatch had been sent and captured that say that Beauregard was on the march to Culpepper?