Switching between low voltage recording of biologic signal, to high voltage stimulation
I'm wondering if I could use an analog switch, to switch between the
two inputs.

Well, the ADS1299 uses a MUX (a type of analoge switch) internally: -

And that MUX interrupts the input lines to inject alternative signals so, the precedent is already set.

I'm not sure if an analog multiplex would work. I'm
worried about noise, since the signals are so small before the
amplifier.

Try it out is my advice.
Or, link/daisy-chain another ADS1299 to handle the simulated signals. That should not cause any additional degradation to the signals: -