Using external microphone disables internal speaker
Since I posted this question, I got a new external monitor that connects to the DisplayPort (Acer EB321HQU). It has its own speaker, so the Sound menu allows me to choose between Internal Speakers and the monitor.
When I plug in the microphone, the Internal Speakers choice changes to Headphone, so it still disables the built-in speakers. But the option to use the monitor as the output device is still available. So I can select that and get audio without a headphone. Plugging in the microphone automatically changes the selection from the monitor to Headphone, but I can easily switch it.
This solution won't work more generally, it's only for configurations with a monitor that can be used as a sound output device.