From the Codex page for wp_insert_post():
IMPORTANT: Setting a value for $post[‘ID’] WILL NOT create a post with
that ID number. Setting this value will cause the function to update
the post with that ID number with the other values specified in $post.
In short, to insert a new post, $post[‘ID’] must be blank or not set
at all.
You just need to supply your feed post with a matching WordPress id and the function will update instead of insert. Unfortunately, I don’t know how the rest of your system works so providing more detail, much less code, isn’t really possible.