Hurrah! We were able to rewrite the app and change the model from stringList to object. As a result, our users can now search multilevel select custom fields in Issue Navigator using JQL. The field values also export correctly (previously it was just an option ID).
Check the latest version of our app: https://marketplace.atlassian.com/apps/1224654/translated-select-for-jira?hosting=cloud&tab=overview
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