The org i work in doesnt use lighnigh so can't help you there but I considered implementing this in classic (but didn't). Below may help you.
Docusign Support - Docusign Javascript Template
With the above template you can create your own javascript docusign button.
The variable that passes contact information to Docusign is the CRLC variable (found in the template above).
Docusign Support - Customising Contacts <- more info here.
to map the fields in the javascript button you will need to have a Contact lookup field on the quote(one for every recipient).
- Employee selects a contact and populates the lookup
- Javascript button uses the relationship field to get the variables from the contact.4
FirstName~contact_lookup__r.first_name
I dont think this is too much work. as long as your comfortable completing the template or adding the CRL variable to your own docusign.