To export a project through Gitlab’s API, first create a Personal Access Token (PAT) and give api and repository permissions.
Export PAT as an environment variable.
Then find the project id (main page of the project – 1111).
To start an export use
curl --request POST --header "PRIVATE-TOKEN: $PAT" "https://your.gitlab.repo/api/v4/projects/1111/export" --data "upload[http_method]=PUT"
Then check the status and get the download link when it has finished
curl --header "PRIVATE-TOKEN: $PAT" "https://your.gitlab.repo/api/v4/projects/1111/export"
{"id":1111,"description":"Project description","name":"project","name_with_namespace":"MyOrg / project","path":"project","path_with_namespace":"myorg/project","created_at":"2021-11-01T10:57:23.195+02:00","export_status":"finished","_links":{"api_url":"https://your.gitlab.repo/api/v4/projects/1111/export/download","web_url":"https://your.gitlab.repo/myorg/project/download_export"}}