You might check the Blu-ray player's manufacturer's website to see if there is a firmware update. There might also be some setting on the Blu-ray player to change how it handles video from over the network.
If you have the disk space, you could try
using the DownloadHelper plugin in Firefox to download the video from YouTube and then play it back in VLC. You might need to edit the resulting .flv file to cut it into manageable chunks.
Is there a way to connect your computer directly to the TV? If your computer can be connected to an HDMI cable, there are inexpensive HDMI splitter boxes which automatically switch to the active HDMI connection, so your setup would look more like this:
TV --- HDMI splitter <