I recently won an NSF mathematical Science Postdoctoral Research Fellowship at University 1. At this University, the math NSF postdocs have a have a much lower salary and benefits than regular "named postdocs" (XYZ Assistant Professor)and during their 2 years of half-time NSF support teach two thirds the amount (despite only half the salary coming from the university). I did not get a regular postdoc offer from Uni 1.
At the same time, I have a 3 year postdoc offer from a fancy institute at US University 2, with significantly higher salary than regular postdocs at Uni 1 (which is already much higher than NSF postdocs at Uni 1), more travel funding than the NSF MSPRF, and NO teaching unless I want to.
Research wise, Uni 1 is a better fit. Would it be reasonable for me to leverage the Uni 2 offer to perhaps convince Uni 1 to raise my salary during my NSF postdoc there to the level of their regular postdocs and maybe give me the title of a regular postdoc at Uni 1 for one or two years when I am there? Or perhaps reduction in the number of courses I have to teach? If so, does one go about it?