I think fixing all affected calls to socket.close in the world (option 3) would be too much. I just added two new reports (Issue 30652 and Issue 30391) as dependencies. These are about testing socketserver.TCPServer. As an example, to fix the socket.close call there, I think the change would look like

class TCPServer:
    def close_request(self, request):
        try:
            request.close()
        except ConnectionError:
            # Suppress asynchronous errors such as ECONNRESET on Free BSD
            pass

Instead of that change all over the place, I am thinking option 2 would be safest. In Modules/socketmodule.c, something like

sock_close(PySocketSockObject *s)
{
    Py_BEGIN_ALLOW_THREADS
    res = SOCKETCLOSE(fd);
    Py_END_ALLOW_THREADS
    /* ECONNRESET can occur on Free BSD */
    if (res < 0 && errno != ECONNRESET) {
        return s->errorhandler();
    }
}