Is the Beyoux Tapestry a reliable tapestry why or why not?

First of all it may be called the "Bayeux Tapestry" but it is in fact embroidered and so is not a tapestry at all. It was commisioned by William's half-brother Bishop Odo several years after the battle of Hastings..
As a historical document it is confused (having many authors), it was almost certainly made in England by Saxon women, the connection to Bayeux comes from it's discovery there in the 18th century.
Although it is purported to display the events around the death of King Harold, there is no way to know which figure is him, there is a figure with an arrow in his eye under the name of the King (this is probably where the notion that he was killed thus came from), but there are a few other figures being killed under his name also.
Remember that the "tapestry" was commissioned by the victors and in these instances it is quite usual for them to claim a greater margin of victory or to use it as propaganda.