Actually they did have a serious problem during the war;
there was an entire illicit trade, even:
Gen. Ulysses S. Grant took command of the Department of Tennessee on Oct. 25, 1862, and immediately started planning to capture Vicksburg, Miss., the principal remaining Confederate stronghold on the Mississippi River. In doing so, he came face to face with the dangerous consequences of a surprising fact of the Civil War: the semi-illicit trade in cotton between North and South. Northern merchants paid well, and often in gold, for the Southern crop; the Confederates then used that gold to buy new weapons, sometimes from clandestine suppliers in Northern or occupied cities like Cincinnati and Memphis. Grant soon found himself confronting rebel troops sustained by such trade, including cavalry armed with modern breech-loading carbines.
It’s well known that British textile manufacturers relied heavily on Southern cotton for their raw materials, but the New England mills ranked second. Consequently, by the autumn of 1862 Northern “cotton speculators” were regularly infiltrating front lines seeking to acquire feedstock needed in Britain and New England. The year before Lincoln’s election, the South accounted for 70 percent of American exports, the great majority of it cotton.
The war, and the federal blockade of Southern ports, sharply curtailed those exports. This wasn’t just a problem for the textile industry: President Lincoln could scarcely hope for a favorable trade balance without cotton. Given that gold was the international settlements standard, the situation could strain the Treasury thereby eroding America’s international status and inviting foreign recognition of the Confederacy. At the same time, Lincoln, Seward and others believed that Europeans might intervene if their textiles industries were overly deprived of the commodity. And so the decision was made to quietly allow intersectional trade in cotton.
As for measures the C.S.A. can take, there are many, given it would be impossible to hide any large scale trade for long. This is also another point in terms of what others have pointed out earlier in the thread, in terms of the loss of the Confederacy changing the economic picture of North America. With regards to Mexico, the Mexican Empire lacked much of a basis without foreign support and had little ability to project power against the Northern Mexican strongmen. Vidaurri, for example, wasn't ousted until late 1863/early 1864 via French arms. With a peaceful annexation, there isn't much a basis for any outside intervention.