Doing this with email-based signing works flawlessly:

            /api/v1/templates/${templateId}/createdocument with quicksend=true

      But I'm not sure if it support or there is an equivalent for embedding

Option 2 - GOOD CASE - Automatic fields

Push the pdf with field tags via API => Send prefilled fields & Generate signing URL for embedding via API

      Trying a document creation with tags works for signer 1
            /api/v1/requests       But I can't find how to do the equivalent for the "prefilled by you" recipient, as we need to prefill some data.

Ideally, option 1 would be supported, as we don't have to upload a file each time (traffic costs + latency), and can configure the fields much faster with the UI, but option 2 would be a decent solution, there is still the option to do it completely manually, as in, without automatic fields or templates, but that one will be very hard to maintain.