For a good read on the subject, I'd suggest
Mexican Projects of the Confederates by J. Fred Rippy,
The Southwestern Historical Quarterly, Apr., 1919, Vol. 22, No. 4 (Apr., 1919), pp. 291-317; It's available on JSTOR if you have access to that. Beyond giving early coverage to the Vidaurri offer, which R. Curtis Taylor would return to in detail decades later with
Santiago Vidaurri and the Confederacy (
The Americas, Vol. 26, No. 1 (Jul., 1969), pp. 66-76), Rippy also covers other fascinating adventures, in particular overtures to Terrazas of Chihuahua and Pesqueira of Sonora, fellow strongmen in the same vein as Vidaurri. Further, planning was afoot at least as early 1863 for many Confederates to settle into Mexico.
All these things together make me suspect that the Confederacy could, at the least, eventually annex Northern Mexico and puppet the rest; personally, and I have advocated it before, I think they could possibly gain the whole of the nation. If the local strongmen are aligned with the Confederates and the aforementioned settlers control the economy and military, that is a very stable basis from which to absorb the country. In the interim, when France is inevitably forced to retreat due to events in Europe, the C.S.A could establish itself as the patron of Mexico and then move to intervene when the Emperor dies. Historically, he was in poor health and was likely infertile, having produced no heirs.