The article simply states that the writer's "mole in the College of Arms certainly thinks" that Archie will be named as Lord Archie on legal documents.
However, he is not named as Lord (or Earl) on his birth certificate, which to my knowledge is the only published legal document in which Archie is mentioned.
Courtesy titles of children of peers are used in British passports, see Annex B in the link below. Archie's should be too.
https://assets.publishing.service.g...t_data/file/118562/observations-passports.pdf
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