Hi scavendish,
Welcome to the Apple Community,
where some of us do our best to help each other troubleshoot.
Guess you have checked that the headset is properly charged, and turned on.
Wondering if you have checked with the manufacturer that your headset is compatible with iPhone 13
https://www.poly.com/au/en/search#q=Plantronics%20headset&t=All
Apple support have some articles online that may be helpful.
Set up and use Bluetooth accessories on iPhone (iOS15)
https://support.apple.com/en-au/guide/iphone/iph3c50f191/ios
Use bluetooth and Wi-Fi in Control Centre
https://support.apple.com/en-au/HT208086
Resolve Wi-Fi and Bluetooth issues caused by wireless interference:
https://support.apple.com/en-us/HT201542
If your phone isn’t working right, try re-starting it:
https://support.apple.com/en-au/guide/iphone/iph841379c3d/ios
iPhone Support:
https://support.apple.com/iphone
Hope there's something here that helps answer / solve the situation :-)