Secretary of State Gresham initially was belligerent to Spain until he got praise from jingoists (who he hated), when he then dismissed the case.
Removing Secretary of State Gresham is easy - he was a low-tariff Republican judge who declined almost every possible career advancement. The only reason he was even appointed was that Cleveland recognized that economically he had been a Democrat for many years, but Gresham only left after Harrison was elected (the two hated each other). He had no FP experience at all, and was a bungler in the role (he dithered extensively in Hawaii and Venezuela; Gresham was an expert ditherer).
Take Gresham out of Foggy Bottom and turn that miss into a hit... well...
For one thing Cleveland wouldn't annex the Phillippines, which means they become German or Japanese most likely.