You want to add the "default" value if it is not there.
Here's a snippet of the JSON for our Google Drive LTI - which is the last one that I modified to disable by default (I've removed some of the values with URLs):
"course_navigation": {
"enabled": true,
"text": "Google Drive",
"default": "disabled",
"label": "Google Drive",
"selection_width": 800,
"selection_height": 400,
},
So in Postman build off the API URL of your external tool (ours are are installed at the account level), so
api/v1/accounts/[account_ID]/external_tools/[external_tool_id]
Key: course_navigation[default]
Value: disabled