But here the question concerns the "German Empire" (whatever that may be in this timeline), and not the Prussian throne. Say that Otto acts way differently in 1848 and becomes a leader of a revolutionary nationalistic united Germany, where he after some time does a Napoleon III and is crowned emperor.
And the OP only wants Bismarck's policy regarding the US. Perhaps we need to know more about what actually led to his reign first.
Ok so ...
In March 1848, crowds of people gathered in Berlin to present their demands in an "address to the king" the main ideology of this movement is that the monarch is only a figure head, with the Chancellor being the head of State and Government.
King Frederick William IV, taken by surprise, yielded verbally to all the demonstrators' demands, including parliamentary elections, a constitution, and freedom of the press. He promised that "Prussia was to be merged forthwith into Germany."
Otto von Bismarck, even at the young age of 32, is swept up by the protesters who see him as their figure head, he declares that if he is elected as Chancellor of Germany, he will bring this nation into the golden age that it deserves.
In 1850, he won a seat in parliament, but was not able to gain a majority to become chancellor. However he did become Ambassador to France, and within his first year, as ambassador, he witnessed the coup d'état in 1851, which saw, President Louis-Napoleon Bonaparte, take the throne as Emperor Napoleon III.
He would see, the first years of the Empire, in which Napoleon's government imposed censorship and harsh repressive measures against his opponents, some say this lead to, his radical idea.
In 1860, with King Frederick William IV of Prussia, deteariating health, von Bismarck, began making his moves in parliament and on the king's death a year later, pronouced himself, Chancellor of the German Empire, hampering King Wilhelm I's idea of creating a Presidency of Northern German Confederancy.
In this year, Kaiser Otto V, offered his support on the side of United Mexican States along side the United States of America, forming an alliance that would stand strong, especially after America offered to support Germany in Europe, if Germany helped the Union, fight the Confederacy.
The American Civil War started in 1861 and ended in 1863, after Imperial German soldiers were able to support the unionists in the north, while the Imperial navy, blockaded all major docks and shipping ports.
Two years, after being named Chancellor of the German Empire, the authority of the Prussian royal family was completely gone, the royal members were only King and Princes by title, Otto von Bismarck was the true ruler and on June 30th 1865, was offered the title of Kaiser of the German People and accepted it willingly.
Although they had been theoretical enemies in Mexico, France and Germany were able to work closely with each other, after both leads showed respect and held one another in high opinions.
1869 - Kaiser Otto V of the Germans and Emperor Napoleon III of the French.