OK, the succession is reversed in 1865. It WAS the President Pro Temp who was next in line. What is interesting is that the Secretary of State was somehow expected to arrange for another Presidential election? Or so Booth thought, apparently. And he expected electoral chaos with both William Seward and VP Andrew Johnson dead. But I suspect that Booth would have been disappointed, and the Union would have continued with far less chaos than the resulting impeachment of Andrew Johnson ITTL.
(BTW, we'll never know of course, but Booth may have been hoping to kill someone else sitting alongside Lincoln at Ford's Theatre that night. Secretary of War Edwin Stanton was amongst the people Lincoln invited to Ford's Theatre, but he declined. So did Ulysses S. Grant. Plenty of butterflies if Grant had accepted and been killed with Lincoln).
I suspect that Lafayette S Foster would have gotten along far better with the Radicals in Congress over Reconstruction. He would have left no reason for the House to impeach HIM. In fact under Foster, we might have seen the Southern States broken up, recombined and reduced in electoral strength, particularly the anti-Union parts of the South and slave owners treated far more severely.
Since Foster served on the Senate Foreign Relations Committee during the Civil War, he likely knew the Russian Ambassador Count Steckl. Thus, when Steckl approached him and whoever he appointed as Secretary of State with an offer to sell Alaska to the United States, Foster would likely approve of it and because of his Senatorial seniority and connections, would probably be able to shepherd it through ratification with much less trouble than Seward did. He might have been able to avoid the hard feelings and buyers remorse about Alaska that resulted in Congress refusing for the next 20 years to appropriate any money to administer Alaska Territory and might even have been able to work out a settlement of border issues with Canada and British Columbia that could have led to a land grant for a railroad to Alaska when the Southern Pacific and Northern Pacific railroad land grant bill came before Congress.
Foster might well have been elected to a term in his own right, serving until 1872.