Welcome to the forum @tgkrueger
Sorry to hear of this issue. I have only done this one time in the past with the Wyze Headphones connecting to an Echo device assisting another member with almost the same scenario as you and it worked without issue.
I tested connecting an Echo Dot Gen 3 to the Wyze Buds today and I was able to connect them right up and listen to music.
Here are my resources
Wyze Buds (Black not Limited Edition) - Firmware Version 0.03.18
Echo Dot Gen 3 - Software Version 7416571268
Alexa App (on iOS 15.5) 2.2.485084 (Client Version 1.24.208447.0)
As far as process goes
I opened the Alexa App, Browsed to the Echo Dot Device and Clicked on the Add button under Bluetooth devices
With the Wyze Buds in the case and the lid opened, I pressed and held the pair button on the Wyze Buds Case for about 5 seconds until I saw the devices available in the list and selected them
The Wyze Buds showed as connected and I listened to music and gave a number of commands. All worked as expected.
Not sure if how you attempted to connect is any different than the way I explained. But check all devices and make sure they are up to date.
I don’t have an Echo studio to test on.