This plugin act on onAfterDispatch event, where the component have finish it's render.
It's using jQuery to alter the DOM. If it did so after CB then it wouldn't be an issue for the tabs jQuery as the tabs jQuery would've done its necessary processing, but since it's acting on such an early Joomla trigger its jQuery is firing before CBs.
But we can parameterize the plugin to just ignore the H2 tag containing some class names (cbTabNav and/or cbTabNavMenu).
The problem can therefore adjust well.
You're altering the DOM to adjust styling. That's slow and dangerous. All you need to do is add the H2 selector to your templates H1 selector styling and it'll style H2 exactly like H1 without having to alter the DOM. If you alter an element with jQuery bindings then you just broke those bindings. At any rate use something like that at your own risk, but from here on please disable it before opening a support request as it can conflict and cause issues.