I downloaded a Word Bilingual file for review but could not upload it to update the translation.

Bilingual Word files are not meant to be used to update the translation. So there is no way to import them back into Smartcat. For this process, it is possible to export XLIFF files. These XLIFF files can be opened in QA tools, other CAT tools or free editors like POEdit which are better suited than Word for editing.

As an alternative, it is also possible to invite the client's reviewers to the project as Linguist and assign a task of proofreader. In many cases, it could be a better solution because there can be interaction between the reviewers and the translation team. A new Smartcat account just for that customer's project could also be created so that there can't be any issue with data getting shared between customers.