Filemaker Pro web viewer access to Dropbox via API and "insert from URL" command

There isn't a way to have the API itself automatically perform logic and additional calls for you like this. The /2/sharing/create_shared_link_with_settings and /2/sharing/list_shared_links routes are distinct endpoints on the Dropbox API, and each require their own request from the API client (i.e., the code calling the API, such as your FileMaker Pro code in this case). You'll need to write your code with whatever logic, calls, and processing are needed for your use case.

It is worth noting that in some cases, the Dropbox API will return the information for the existing shared link in the 'shared_link_already_exists' error itself from the /2/sharing/create_shared_link_with_settings endpoint to begin with, but this isn't guaranteed to always be included, so you would still need to implement the code to call /2/sharing/list_shared_links anyway, for when it isn't.