Diagnosis
The Chime Pro is successfully connected to your Wi-Fi, but it is not providing an adequate extended signal to your doorbell. Placement is the most common cause.
How to Fix It
Check the RSSI value for both devices in the Ring app. Go to Device Health. For the Chime Pro, the RSSI to your router should be better than -60. For the doorbell, the RSSI to the Chime Pro should also be better than -60. Physically move the Chime Pro. It must be positioned halfway between your router and your doorbell, not right next to either one.
If That Didn't Work
The Chime Pro may be extending a weak source signal. Temporarily move the Chime Pro to an outlet right next to your router. Set it up there. Once connected, move it to its intended midpoint location. It will reconnect to the same network from the new spot.
Heads Up
The Chime Pro creates its own SSID (network name) for Ring devices. Your phone will not see this network. The doorbell's connection to it is managed automatically by the app after initial setup.