This extension allows to get language labels from ext:translatr as JSON.
Use composer:
composer require sourcebroker/t3apitranslatr
After installing the labels are accessible by following endpoint:
/_api/translations
To narrow the number of labels fetch you can use tags on ext:translator and get only those tags that are needed on page.
/_api/translations?tags[]=general&tags[]=user
See https://github.com/sourcebroker/t3apitranslatr/blob/master/CHANGELOG.rst