Lincoln had no particular problems working with Chase at the beginning of his first term. Chase, unlike Cameron at the War Department, was energetic and competent as could be expected in such an unprecedented situation. And, unlike Seward at the State Department, Chase had no expectations of being the power behind the throne nor did he try to act on his own in "negotiations" with the various states.
What did happen was that Chase, who was a harder Abolitionist than Lincoln, was more popular with party leaders and citizens who favored much more movement towards abolition, plus other complaints, particularly as to competency. Chase tried to organize a campaign for the Republican nomination for the wartime election. Lincoln got wind of it and timed a demand for public support for the administration by the entire cabinet. Chase, who essentially agreed with all other policies, was then forced by principle to undercut the supposed ground of his own candidacy. Lincoln later got rid of Chase in the second term but moved him into the Supreme Court. (Where Chase ultimately ruled that the greenbacks he issued as Treasury Secretary were unconsitutional.
)
Lincoln had a serious personal problem however with Chase's daughter, who was married to a Senator Sprague. She apparently had the temerity to lecture him. As I recall he called her "quite a politician," which was
not complimentary.
As to whether Lincoln deliberately picked political rivals to "keep an eye on them," there was probably a relative dearth of experienced Republican leaders. The ones who came from other parties originally had support outside the fledgling party. Which would mean it would have been useful to gain that support by offering prestigious jobs to the leaders. Lincoln was an honest politician who would rather jump out a window than tell a lie, but he missed very few tricks as a politician. I'm sure the possible political benefits were also a consideration.
Seward had a mystifying reputation as a firebrand Abolitionist, which his presidential campaing and later events showed. Chase was a longterm member of the Liberty Party, a minor third party dedicated to abolition. Later he was a founder of the Free Soil party, along with Van Buren democrats in New York. In addition to that political disagreement, Seward was a New York leader of the Whig party, closely associated with Thurlow Weed, a figure widely regarded as a machine politician of dubious integrity. Chase was a devout evangelical Christian while Seward was a worldly man who put state funds into Catholic parochial schools in New York. Seward seemed to do nothing but talk while Chase's work in financing the war was unsung but vital for victory. Plus, Seward was the man who got the prestige and was regarded as Presidential while Chase was not, though Chase wanted to be. It would have been surprising if these two had gotten along.
