That seems like an undercount for Pemberton at Champion Hill. Going by memory, Stevenson's division was about 10,000, Bowen's about 4,000, Loring about 7,200, Wirt Adams 800, 21,800 total. Vaughn guarding the railroad bridge 8 miles back.
If we subtract Reynolds being sent away to escort the supply train, that's still a little over 19,000.
If you're just talking about the forces initially engaged, that's Hovey+Logan (12,000) vs Barton+Lee+Cummings+Wirt Adams (8,100).
It is true that Pemberton, like other Confederate commanders, was certainly giving in his narrative reports his "force", "strength" etc. as his total
"effective" force on hand, which generally excluded officers, and regarded only men in the rank and file, armed, in line considered fit for combat; irrespective of the actual number present for duty, which included numbers of men detailed, detached, sick, unarmed, straggling, etc. etc.
Col. William Allen, late of the CS Army, observed that the present for duty strength on the returns, included numbers of men attached to the trains, other than the "effective" combat force/strength...
And that the term "effective" relative to strength, force, etc. regards only enlisted men, among those present for duty, with their units...
So we see as a general estimate, that Pemberton's commands employed in the march to Champion's Hill, included something like 22,000 all told. But not all were "effectives" which were described as somewhat less, about 17-18,000.
General S.D. Lee suggests just so post-war, in the 1893 speech unveiling the Vicksburg defender's monument, whatever the total number of officers and men with Pemberton's army on May 16, it included about 17,000
effectives...
As did General Johnston in an 1870 letter..., quoting Pemberton as to 17,500 effectives...
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...
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And that of the over 22,000 troops of all ranks which were as likely as not marching with Pemberton, a certain proportion were officers, and among the men were not effectives, or though present for duty, were detailed or employed outside the ranks, as with the baggage, besides the sick, stragglers, etc. How many exactly is not recorded. Pemberton's trains by May 15-16, I've seen described as large, and to number in the hundreds of wagons.
On that score, Pemberton himself stated (apparently excluding officers and ineffective, etc.) as of the 14th of May, he was actively maneuvering a
force of "about 16,000"
excluding Vaughn and Tilghman with about 3,000 between them. And by appearances he expected significant straggling in the next days threatened to reduce that
force...
That night, he reported he would march toward Dillons on the 15th once he gathered a column of 17,000 "
men..." (apparently excluding officers)...
Which march he commenced about 1:00 P.M. on the 15th (after some delay), with strict orders to prevent straggling which might reduce that force...
General Pemberton reported on May 17, his whole "disposable
force" in these movements was about 17,500...
James H. Wilson's journal, on General Grant's staff, reports from intelligence, that the aggregate of Pemberton's force which had marched on May 13 was estimated at 25,000 aggregate with at least 20,000 soldiers reaching Edward's by May 14:
Of which force, as Pemberton noted, he formed on the 15th a column of ca. 17,000 effective men.
Whatever total number of officers and men with Pemberton's command were marching to or about the battlefield on the 16th, in the ranks, with the trains, or straggling, etc. which as you mention was likely, all told, in excess of 20K officers and men; not all of the troops of Pemberton's army were engaged in the battle that day for the customary reasons.
Among them Captain Wilson of the 2nd Missouri...
Pemberton's aide-de-camp J.C. Taylor noted spending much of the action on May 16 attempting "pushing up" stragglers to the field of battle...
Gen. J.H. Taylor, with Pemberton, noted straggling
from the field of battle at the close of the action particularly...
Pvt. John G. Earnest of the 60th Tennessee Volunteers posted in the rear with Vaughn's brigade, noted as the mass of stragglers from the battle moved rearward, so did the large Trains of the Army (with all of the men attached to it)...
"About ten o'clock A.M. the firing was terrific on our left and continued until about three in the evening, prisoners coming in all the time. At this time the stragglers commenced arriving and then came the wagon train in a perfect rush. I knew what was up. We were falling back--but in pretty good order. Our company was posted on the rail road to stop stragglers but they came so fast we had quite a time of it. At length the enemy came right up in sight when one of General Pemberton's staff ordered us back to Big Black Bridge--here we were posted for the same purpose as the evening before. Remained on duty all night."
Just as General Grant stated he was certain not more than 15,000 of his own troops were seriously engaged on May 16 (excluding most of McClernand's corps, much of which was involved in more or less skirmishing before reaching the scene of the general action at its close); General Pemberton, in a post-war explanation, excluded from his engaged force General Loring's Division (which faced McClernand, etc. and broke off to join Johnston's army)... and calculated less than 11,000 of the men of Stevenson's and Bowen's Divisions, etc. were
in the fighting...
Presuming in the above Pemberton is giving just the effective force, excluding officers, from those in the fighting (less than 11K), per Livermore (p. 99) recommending addition of about 8 percent for officers, so by that probably about the better part of a thousand, or ca. 12K officers and men in the actual fighting.
Bringing the