I would venture to suggest (and unlikely as it may seem thinking of the pictures of the elderly pipesmoking academic) Tolkien was a more than competent schoolboy rugger player (first XV if i remember rightly) so I think it almost certain that it wolu have been a flabby rugby football..... not an Association one
I am sure a dwarf scrum would be fearsome but might be lacking speed in the back five - I think you really would want elves there,
In a middle earth team I would be thinking in terms of a Dwarvish front row, Men of Numenor would be great in the second row and I am sure that Elladan and Elrohir would make a fine pair of Flankers..... with maybe a Ranger of the north at No 8 - Hoewver the Free peoples might struggle against the Dark forces XV, given their access to cavetrolls, Ologs & Uruks ...... a formidable pack but lacking in wit and intelligence in the line out and set pieces?
It has to be said that almost all of middle earth seems to have an inexplicable aversion to my favourite sport of swimming..... it almost always seem to end in tears doesn't it?... drowning, losing/finding rings of power,getting shot by orcs, nearly losing your barrel, falling into enchanted sleep........