I really hope you do not believe what you are saying here. This is ten days after the Battle of Antietam and you are claiming a decision on spending by Halleck is an order to abandon the pursuit.
On the
18th, Lee is still on the field.
On the
19th, McClellan pursues to the water's edge.
20th, he tries crossing 5th Corps to hold a debouche over the Potomac. Lee responds in great force (about a quarter of his army, i.e. 10 brigades) and stops him.
So McClellan can't pursue over the Potomac at Shepherdstown, because Lee is not sufficiently a beaten enemy. Lee can respond and engage at Shepherdstown, so it is not safe to cross there.
What McClellan can do, however, is that he can continue pressing Lee by crossing the Potomac somewhere else - specifically, at Harpers Ferry. On the 20th Harpers Ferry is reoccupied by 12th Corps.
On the
21st there is some question about whether Lee is planning to recross the Potomac to the north (at Williamsport) and McClellan can't send most of his army marching down to Harpers Ferry (which is further from Williamsport, obviously) because Lee may be about to cross into Maryland again. So on the 21st McClellan sends 6th Corps up to Williamsport to join Couch's division.
Late on the 21st, McClellan orders 2nd Corps down to Harpers Ferry, and Sumner marches as directed at dawn on the 22nd. (McClellan also orders 1st Corps down there, but this order is then countermanded.)
22nd, 2nd Corps reaches Harpers Ferry and Sumner is ordered to build a pontoon bridge there (the rebels had destroyed the bridge). So McClellan is engaged in measures to allow him to cross the Potomac with a complete force (with wagons) somewhere that Lee cannot stop him; note that if Lee does move to block Sumner's crossing then Franklin can cross unmolested at the Williamsport fords.
Also on this date, McClellan asks for crews to repair the rail bridge, and asks Halleck for the expense to be authorized.
23rd, McClellan sends 9th Corps and most of his cavalry down to Harpers Ferry. Halleck asks for McClellan's plans.
Also on this date, the pontoon bridge at Harpers Ferry is laid, but a storm destroys the bridge before it can be used.
24th, McClellan replies to Halleck explaining his plans. If the river rises enough that it's unfordable, he intends to concentrate most of the army at Harpers Ferry and cross into the Shenandoah towards Winchester.
Obviously crossing the Shenandoah under these conditions requires a bridge to be built there - either a wagon bridge or a rail bridge that can resist freshets.
It's these appropriations which Halleck refuses, at the same time as arguing that McClellan should go into Loudoun Valley instead of into the Shenandoah.
The fact that the Potomac is between McClellan's command and Lee's, and that Lee was willing (and, apparently, able) to court engagement near the Potomac to block McClellan's crossing attempts means that a simple pursuit to "keep up the scare" is not sufficient. McClellan functionally needs to launch a new campaign against the enemy, and he needs to do so from a crossing Lee cannot block; that being Harpers Ferry.
Since McClellan is operating from Harpers Ferry, though, he will be operating a few days from his base already by the time he reaches his target (which is Winchester to cut Lee off), and so he needs a solid crossing that can't be interrupted by a storm (which would cut him off from the North and from his supply lines). It is this which Halleck's refusal prevents.
So what prevented McClellan's pursuit? Take your pick - if only a move over the Potomac directly following Lee counts, then it's the fact Lee was willing to throw a corps at preventing the crossing there. If crossing at Harpers Ferry counts, then it's the storms and Halleck not wanting to let McClellan build a bridge the storms can't destroy.
There is perhaps an argument to be made that McClellan should have shifted 2nd Corps to Harpers Ferry a day or so earlier, and got the pontoon train into action there a day or so earlier (which may or may not be feasible depending on where it is); beyond that though a lot of the possible time savings come down to either knowing what Lee's intent is before Lee does (i.e. knowing that Lee is not actually going to try and cross at Williamsport) or bringing on a general engagement at Shepherdstown with the possibility for McClellan to be caught mid-crossing by Lee's entire force.
Something that is worth considering at the same time, by the way, is that Halleck is repeatedly reluctant to agree to McClellan's "Shenandoah plan", and the cause of his reluctance is explicitly the need to keep Washington safe and the risk of an army occupying Manassas while McClellan is in the Shenandoah.