Per the help page above:
“Putting a website into Compatibility Mode will:
Disable content blocking
Disable tracking prevention
Disable web fonts blocking
Enable cookies
Use the Safari user agent
Disable all browser extensions
”
If you disable compatibility mode, and set the per-site useragent for Gmail to safari, you should be able to use your extension, and it should otherwise behave identically.
If you still have an issue, make a new profile without any extensions and ensure Gmail works (without compatibility mode, without extensions), then add the extensions one by one and that should help find the culprit.