There is currently no single API call to get the HTML source for the template, as in API 2.0. Below is the answer I received from MailChimp support on December 9, 2016:
Thank you for contacting us with your question. Currently, API 3.0 does not contain a direct effect that allows you to restore the "source" of the template. However, we continue to update API 3.0 in the hope of achieving parity with 2.0, so I will pass this feedback on to our team. At the same time, the only workaround available for extracting HTML in the same format would be to create a Campaign using the appropriate template, and then extract the source using the endpoint / campaign / {campaign_id} / content.
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