In ArcMap or ArcGIS Pro, you can use the search_query parameter in the Add Locations or Calculate Location tool to force the points to locate on an edge with a specific ID from some field.
Add Locations—ArcGIS Pro | ArcGIS Desktop
I don't think the REST API supports a search query, but if you're pre-calculating your location fields anyway, you could probably just do it this way.
Sign in to post, follow content, and more. New here? Register for free.