You probably close the stream (try-with-resource) when you throw an exception. Try to define a separate stream and wrap it with StreamingResponseBody and then the exception will be handled correctly @ControllerAdvice.

StreamingResponseBody responseBody = outputStream -> {
    try (ByteArrayOutputStream byteArrayOutputStream = new ByteArrayOutputStream()) {
        //some logic here
        byteArrayOutputStream.writeTo(outputStream);
    } catch (IOException e) {
        //byteArrayOutputStream will be closed but not outputStream
        log.error("Error occurred while streaming: {}", e.getMessage(), e);
        throw new CustomException("Error due to I/O issues.", e);
    }
}