If you're going to resort to Constitutional arguments, you shouldn't gloss over the fact that the federal government is supposed to be one of enumerated powers, and there's no 'bribing universities to do what you want' federal power.
It depends on your understanding of Article 1 Section 8:
>"The Congress shall have Power To lay and collect Taxes, Duties, Imposts and Excises, to pay the Debts and provide for the common Defence and general Welfare of the United States; but all Duties, Imposts and Excises shall be uniform throughout the United States;"
What does "general Welfare" mean in this context? Are those words just meaningless filler, or should they be interpreted to indicate that the spending must be in furtherance of another specifically enumerated power? I believe the latter (Madisonian take), but this is a contentious subject:
https://en.wikipedia.org/wiki/Taxing_and_Spending_Clause#Gen...
Unfortunately, that's not true. Article 1 gives congress very broad budgetary powers. Basically congress can spend money how they want, including bribing universities.