Hmm, that's an unfortunate situation indeed (Twitter does not allow to select which type of tweets to auto-share on Facebook). However, let's try another (possibly) elegant solution this time, as follows:
First off, please open up Facebook and make sure that you are switched to your personal profile (and not your Facebook Page.)
Next, open up the following link:https://apps.facebook.com/youtube-app/
Click "Install Application!" and then select the Facebook Page to install the app for. Click "Add Page Tab".
Next, you will see an "Youtube" tab added at the top of your Facebook Page. Click on that tab, and then click "Application Settings" under that tab.
Now, just fill in your Youtube account name and click "Save settings".
ALL of your videos for that account should now show up below. This will allow you to keep all of your Youtube content in a separate tab on your Facebook page, and you will just need to update your users to check that out whenever you post a new video on Youtube.
Please try this now, and let me know if it works for you. Thank you.