You can "export a rigged puppet" using the "Export Puppet" from the menus, but I don't think that is what you want.
You want to create a short scene in Ch with that puppet in it, then export that clip into Prem Pro (with a transparent background). You can do this with dynamic linking (or by exporting a video clip with an alpha channel). For dynamic linking, create a new scene in Ch, set the duration property to 3.5 seconds, do your animation of the mouth. Then open up Premier Pro while Ch is open, and drag the Scene object over into Prem Pro. This will create a "dynamic link" between the two apps. If you update the Ch scene (e.g. scale the character a bit smaller), it will automatically update in Prem Pro.
Note: I normally expert a video clip with transparency instead - I don't use dynamic link very often. But should work if you have updated Ch and Prem Pro to a compatible version (e.g. the latest version for both). I export a video clip using a video format that supports transparency (e.g. NOT mp4!!) because I like having a video clip file. Nice, robust, safe. But every time you update the scene, you need to export the clip again - dynamic link makes that faster to do and should work pretty nicely with a single puppet and a short scene.