Origins and history for phrase "tote that barge"?
Solution 1:
Merriam-Webster's Eleventh Collegiate Dictionary (2003) indicate the range of actions that tote can apply to:
tote vt {prob fr. an English-based creole; akin to Gullah & Krio tot to carry, of Bantu origin; akin to Kikongo -tota to pick up, Kimbundu -tuta to carry} (1677) 1 : to carry by hand : bear on the person : LUG, PACK : HAUL, CONVEY
The original meaning of "carry by hand" is the only one listed for tote in Merriam-Webster's American Dictionary of the English Language (1852):
TOTE v. t. To carry or bear {A word used in slaveholding countries ; said to have been introduced by the blacks. ... It is most used in the Southern and Middle United States, is occasionally heard in New England, and is said also to be used in England.}
But Webster's New University Pronouncing Dictionary (1856) includes the more-general second definition as well:
TOTE v. t. To carry or convey {Local.}
An example of toting in the sense of "conveying"—or more precisely, "towing" appears in Elbert Hubbard, "The Gentle Art of Defamation," in the Washington [D.C.] Herald (October 11, 1914) in a discussion of the hazards of transporting poorly manufactured gunpowder on board ships:
France has learned a bitter lesson in the line of manufacturing powder for itself. To date it has lost three battle ships.
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And so we have the peculiar spectacle of a French battle ship going out for target practice, toting behind it a barge on which its powder was stored, this because the sailors would not take any chances. The powder was then carried on board, in small quantities and fired without delay. And what was left after the target practice was carefully carried back and deposited on the barge, which was pulled by a hawser of a generous length.
The Wikipedia entry for Towpath indicates that people sometimes pulled boats along waterways:
A towpath is a road or trail on the bank of a river, canal, or other inland waterway. The purpose of a towpath is to allow a land vehicle, beasts of burden, or a team of human pullers to tow a boat, often a barge. This mode of transport was common where sailing was impractical due to tunnels and bridges, unfavourable winds, or the narrowness of the channel.
The Free Dictionary, drawing its information from The Great Soviet Encyclopedia (1979) reports that human "barge haulers" were employed in Russia from the late 1500s forward:
Barge Haulers
river workers who towed boats by means of ropes along the river bank or on the water by means of oars. The first barge haulers appeared in the late 16th and early 17th centuries. Barge hauling expanded as a consequence of increased river transport at a time when the technical development of transportation by water was very imperfect.
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Barge hauling declined in importance as technological advances were made. At the beginning of the 19th century there were as many as 600,000 barge haulers working along the Volga and Oka rivers; in 1851 there were about 150,000. As shipping developed, the number of barge haulers declined still further. By the beginning of the 20th century barge hauling had disappeared.
As phoog notes in a comment below aparente001's answer, Ilya Repin depicts Russian barge haulers in his painting Barge Haulers on the Volga, completed in 1873. It isn't at all difficult to imagine slaves in the United States performing the same laborious task as Repin's burlaki.
Solution 2:
Mules would pull barges down a canal. Maybe (shudder) people were also used to pull barges along a canal.
Solution 3:
Here is an example from 1814 where it means to tow (by horse).
Perambulations of Cosmopolite: Or, Travels and Labors of Lorenzo Dow,
I came to a camp where some negroes were toting* tobacco to market. I stopped with them until day, and one gave me some corn for my horse.
*The mode of toting tobacco to market, is by rolling it in casks, with a wooden axle through the midst, on the ends of which are fastened the shafts for the horse to draw it by.
Four Volumes of Lorenzo's Journal Concentrated Into One By Lorenzo Dow January 1, 1814