So as I understand things you had a V6 box that you have now upgraded to a 360 and the V6 and projector had sound. Since the upgrade to 360 you now have no sound, have you checked if the 360 and HDMI cable with a TV to make sure that combination still works.
What is the projector make and model, does it support 4K and HDCP 2.2 as pointed out by newapollo, and is it a high speed HDMI cable.
As you are getting no sound it might be difficult to do the pairing as the test involves changing the volume but does the 360 recognise the projector make and model number and can you do any of the settings again as suggested by newapollo.
Does the projector have more than one HDMI connector that you can try.
Can you change the video out settings on the 360 to see if that makes any difference, bit of a long shot but maybe worth a try.