Sumner wouldn't win; he's far too radical for the GOP, not to mention the electorate at large.
Perhaps the most significant foreign policy divergence from Lincoln's is that he will cast the ACW as a war on slavery from the outset instead of waiting until 1863. This will make the UK and France much more uncomfortable about any Confederate sympathies they may have.
As a side note, Giuseppe Garibaldi might be made head of the Union forces, the main reason he declined the position IOTL is because Lincoln refused to make it a war on slavery from the get-go.
That being said, I'm not sure how well Northern troops will deal with a foreign (and Papist) general in charge of them.