Ok, I took a brief look at controlling the scope with SCPI over USB. Was thinking about using a MAX3421E as the USB host and controlling that from an 8-bit MCU. It would be even cooler to use a WCH CH554 MCU or similar by itself, but I'm not sure how to get the debugger for those MCUs. Then I looked into the USB protocol a bit and realized that learning it would be a big lift for what I want to do here.
It turns out that setting my scope (SDS2000X+) to normal mode and triggering it externally with a foot pedal switch does just what I want (take a new single shot each switch press), but only if the switch is properly debounced. So I put together a little circuit that debounces the switch upon both open->closed and closed->open transitions, sets a latch on switch debounce closed, and resets the latch on switch debounce opened. Every time the latch is set, it fires a ~100µs one shot that controls a ~5V level line driver. I also added an output overvoltage crowbar to hopefully prevent damage to my scope if something fails in the circuit.
Boards are in the mail, and components are ordered. The circuit simulates well, so I'm hopeful for a good result.