2nd Corps
It is worth considering how French's division was formed.
Kimball's 1st Brigade was one of the two brigades that was sent to reinforce McClellan during the Seven Days. The other brigade was assigned to replace one of Casey's. Kimball's brigade was not placed in a division by Sumner, and had arrived without artillery..
The 2nd brigade was formed by four new regiments (14th Ct, 108th NY, 130th and 132nd Pa) joined en route. On 10th September, Sumner issued an order for BG French, who was the senior brigadier in the corps, to assume command of Kimball and the new regiments as a division. The 2 batteries consisting the reserve artillery of the corps were assigned, on the same date, to be French's divisional batteries.
The 3rd Brigade was supposed to be part of the 9th Corps from SC, but only arrived on the 16th, and was added to the division.
Finally, Hazard's B/RI Bty had entered service as Stone's Brigade Battery, and had remained there as Gorman's Brigade Battery in Stone's (later Sedgwick's) Division. Sedgwick's division had fought the Peninsula campaign with three batteries, one attached to each brigade. When French's division was formed on 10th September, Hazard's Battery was transferred to it, leaving Sedgwick with only 2 batteries.
However, as I'm sure you're aware, on the 16th Sumner detached all batteries from their divisions, sending 6 over Antietam Creek, and sending two to defend the crossings of the Potomac the corps would use the next morning. For example,
Tomkins reported that he camped on the far side of the creek and reinforced Hooker around 0800. After the battle the 3 batteries reverted to French's Division, like the others.
12th Corps
Like 2nd, it's worth considering the origin of 12th Corps.
William's Division was Banks' old division, and both batteries had been with the division since October '61, when it was formed.
Bty F, 4th US started the war as the instruction battery for the 4th US Arty. In peacetime, each US Arty Regt was authorised one battery of 4 mounted guns, one instruction battery for fortifications, and the other companies were organised as a standard infantry regiment. The US reverted to this practice postwar. They drew 6 guns from Pennsylvania state stocks, and joined the Dept of Pa as the brigade battery for Thomas' 1st Bde. Thomas was reassigned, and the brigade became Stone's Bde, which in October was designated 3rd Bde, Banks' Division.
By September '62 this brigade was under Gordon, and Bty F, 4th US was still the brigade battery, exactly as it had been since 30th April 1861.
Bty M, 1st NY had similarly been the brigade battery for the 1st Bde since Jan '62.
The other brigade of the division (Candy's) was transferred to the 2nd division after Cedar Mountain, and their brigade battery (Pa F) went with them.
Greene's Division had been organised in May '62 under Sigel, mostly from units that were manning the Washington Defences. The 2nd and 3rd bdes were the original division, which had been reinforced by a transfer from the 1st division in August.
Pennsylvania Battery F had been the brigade battery for the 1st bde since Banks' division formed in 1861, and transferred in with the brigade in August.
When Sigel's Division formed it was assigned two batteries, one each to a brigade. Bty L, 1st NY to 1st Bde, and Bty K, 1st NY to 2nd Bde (later 2nd and 3rd after August). When they moved down towards Cedar Mountain, Geary's independent brigade was absorbed by the division and broken up. Geary's brigade battery was Pennsylvania Bty E, and hence it entered the division.
However, in McClellan's reassignment the two NY btys were temporarily consolidated into a single full strength battery, and assigned to 1st Corps.
To give the division the required no of batteries, McClellan transferred the 4th/6th Maine Bty (temporarily consolidated) from 3rd Corps (remaining at Washington).
We know Pa Bty F was the 1st Bde bty, but the brigade assignments of the other two are not known.
The 10th NY Bty was not with the army on the 17th.