I managed to solve my problem. The issue seemed to come from the example sketch included with Teensyduino found under File > Examples > Teensy > USB_Joystick > Complete. I’m not an experienced arduino-er, but with that sketch the buttons seem to be sourced and updated using a “Joystick.send_now” command. When I modified the sketch to source and update based on the “bounce” method, the buttons worked as expected. The bounce method is used in another example sketch: File > Examples > Teensy > USB_Joystick > Buttons. Modifying that approach according to my setup was fairly simple to get it working.
In the end, this really had nothing to do with RetroPie.