You would need to add the script tag to an aura component then load your LWC inside the aura component also. Also if the script calls out to another source inside it you will need to add them to the trusted sites also.
Static Resources are not subscriber-editable, i.e., cannot be changed in subscriber orgs when installed as part of a managed package.
Static Resources are meant to be just that - static. They are not for dynamic file storage. If your managed package needs to store writeable document blobs you should store them somewhere else, most likely a ContentDocument.