in translation would I add the word funeral or just leave it as pyre? 
Homer describes the pyres “burning”, so it must be considered to be a physical object, rather than an event.
in other words… a way in which to dispose of the dead.
so just PYRE.
right 
yep. 
Hey, ΤΙΜΟΘΕΥΣ.
It depends on the context and the intended audience, is my opinion. Amongst hellenists, pyre should suffice. For the masses (don’t push!), adding funerary as an adjective of pyre (instead of adding a phrase such as ‘for the funeral’) would be a generous touch.
Yes, funeral pyres, definitely. Otherwise the readers might just assume that the Achaeans were very fond of Guy Fawkes’ night.
and just like you Eureka, funeral in Italics