First off, I need to learn to look closer at things. The pump manufacturer says this is a 1.5" outflow, so I bought a 1.5" hose. You'll note that the old hose (below) is attached not to the pump directly, but to an adapter of sorts that I can only assume makes it 1.25". Anyway, with some extra tightening of the screw clamp the new hose fit fine.
This is your sump pump after who knows how many years under water. This is your sump pump hose, cleaved to the adapter even after the clamp is removed.
Unfortunately, upon securing the new clamp, turning the power back on, and resubmerging the pump, it did not... pump. Some light surgery revealed that either moisture had finally gotten in to the power housing and corroded the connections or something was wrong in the motor section which I can't access due to severe corrosion and buildup around the screws holding the main housing.
Long story short, $110 at Lowes got me a new pump and a one way check valve that allowed me to connect the hose to the pump.
Few things make you feel so much like a homeowner as having to say 'Ben, I'm going to be late - I'm up at Lowes buying a new sump pump so my basement doesn't flood.'