Harold Wentworth, American Dialect Dictionary (1944) confirms your surmise that the word chance in this context means "(large) number":
chance, n. ... 2. A quantity or number, usu. in 'a right smart chance,' meaning an indefinitely large one; also, a crop or herd, distance, company; — used also absolutely, & with monosyl[labic] adjs. (smart, right, fine, nice, great, good, whole).
1823 West 1827–37 smart chance. Provincial. Sherwood Gaz. of Ga. 1829–30 s.U.S. 'He lost a right smart chance of blood.' Vulgarism. Dunglison Glossary. 1856 I'm acquainted with a right smart chance of gals in Keokuk. Twain. Before 1883 Va. he expects to raise 'a fine chance of curcumbers.' Bagby Old Va. Gent. 1895 s.e.Ky., e.Tenn., w.N.C. There's a right chance o' snack houses down to Bakervul. 1899 s.e.Va. Warwick Co. He's got a right smart chance of children. Green. 1902 s.Ill. right smart chance of taters or money. 1903 s.e.Mo. There will be a smart chance of peaches this season. 1908 e.Ala., w.Ga. 'He's got a right smart chanct of cotton.' Common in 'right smart chance,' but often in other collocations. 'We caught a nice chance of fish.' 1915 s.w.Va. Scott Co. We made a great chance of apple butter this year. 1925 w.cent. W.Va. 'That is a right smart chance of cattle." ...
The narrator in the song "I'm a Good Ol' Rebel" says that he fought for General Lee, which means that he was enlisted in the Army of Virginia during the U.S. Civil War—and which puts him geographically in the midst of the upper South at the right time to be using chance in the specified way. The only thing not consistent with Wentworth's coverage of chance is that the word appears in the song lyrics without a modifier such as right, smart, or nice. But this usage may be as much a response to requirements of conciseness in song lyrics as anything else.
Update (June 18, 2021): Matches for 'smart chance' in the relevant sense from early newspapers
An Elephind search turns up matches for "chance" in the sense of "number or amount" from as early as 1819 in Illinois. From a letter to the printer of the Edwardsville [Illinois] Spectator (July3, 1819):
There is a heap of talk in this settlement about the name which the commissioners have given to the seat of government that they allow to build in the woods. They call it Vandalia; but I can’t find out where they got the name, or what it means. Some folks allow that Vandalia was a great general, and that he has killed a power of British and Indians; while others reckon that he must be some big man that has made long speeches in congress.—There is a man just moved into our settlement, that we suspicion for a Yankee, he has a powerful chance of books, and maps, and such like truck, besides a pretty smart chance of plunder.
From a letter to the printer of the Edwardsville [Illinois] Spectator (August 28, 1819):
I wish you to announce, for the information of the public, that a smart chance of the laws of the last session have been printed, and that a heap of such as have, are now in town, which, though incomplete as they are, and without fixings, marginal notes, or index, have made the folks here mighty glad.
And from a letter to the printer of the Edwardsville [Illinois] Spectator (July 10, 1821):
I have heard a heap about a big jail to he built in your town, and about taxing the people to pay for it. Reckon, Mr. Printer, you town-folks have a heap of money or expect some great advantage by having so fine a jail—reckon some body live in it to "entertain company"—folks in my neighborhood say you going to have a fine brick house all round a huge log house with a smart chance of Iron barrs and spikes—dont know what for—reckon to make town look fine—reckon soon you’ll want a fine court house—town look mighty fine then—the people pay for it too—no matter, not much out of town-folks pockets—folks say the tame horse bears the burthen—reckon its true.
Other instances of the "chance" used in this sense appear in the 1820s or early 1830s in newspapers from Indiana, New York, and Virginia—but the great majority of early instances are from Illinois and Indiana. This suggests that the U.S. Midwest, rather than the U.S. Upper South, may have been the region where the usage in question first proliferated.