The earliest possible trace of the shanty that I have found is made up of two lines in B. T. Pindle's patriotic poem, "Lines to General Taylor before the Battle of Buena-Vista" (pub. 1851).

Which came first, the shanty or the poem, is unknown, but the lines are surprisingly similar:

"That gallant Taylor's won the day,
And now is safe in Monterey."

Except for praise of Taylor and the mention of Santa Anna's name, that's the only resemblance.