Hello! I understand that you're looking for a way to download a private video that you're unable to download using IDM.
One workaround that you can try is using a screen recorder. You can simply play the video on your screen and record it using a screen recorder software. There are many free screen recorder software available online such as OBS Studio, Bandicam, and ScreenFlow.
Another option is to use a browser extension specifically designed for downloading private videos. One such extension is Video DownloadHelper which can be added to your browser (Firefox or Chrome) and can detect videos playing on any website. It will then enable you to download the video in various formats.
Lastly, you can try using a download manager software other than IDM such as Free Download Manager or JDownloader. These managers might have a better success rate in downloading private videos.
I hope this helps!