There is no need for a "workaround". Simply remove the labels as usual:
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Select one label.
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Type CTRL+A to select all labels (of that type).
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Type DELETE to remove the selected labels.
Anonymize retains the empty labels because the changes applied by this feature should not affect that result of layout algorithms in any way. I.e. if you run a layout algorithm on your diagram before anonymizing and after anonymizing, the result should be the same. Since layout algorithms can take labels into account, Anonymize has to keep the (empty) labels.