This is a known limit of some Bluetooth headsets, not a problem with your setup. Here's why it happens and how to fix it during your call.
Why this happens
When you use type-to-speak, Rylo Desktop sends audio through a virtual microphone called the Rylo Microphone. That's how the other person hears your typed messages.
Some Bluetooth headsets stop picking up sound when they're not the active input in your call. That can happen as soon as the Rylo Microphone takes over.
How to fix it
Switch your input to your computer's built-in microphone, or a wired headset if you have one. Your voice will be captioned again right away.
How do I know if this is happening?
Rylo Desktop shows a banner during calls where type-to-speak is on, to let you know this might happen. That way, you can switch ahead of time.