With our Product Labels & Badges Pro app, you can show the label on product pages/collection pages/other pages.
However, there is no option to exclude from a specific page or block yet. Thus, you can contact us at sales@bsscommerce.com so we can consider customizing the app for your website.
However, it might take an extra fee - based on the complication of the customization requirements because this is not a feature of the app.
Please consider.