A rather late and incomplete response to this general question of whether Hobbit archers did or did not attend a certain battle.
Whenver I read this, I think automatically about the contribution which Canadian soldiers made on the D-Day, June 6th assault on the beaches of Nazi-held France. Yes, yes, indeed. The Canadians had their own beach, Juno beach. But you know what? That wasn't as big a newstory as the beaches taken by the Brits or the Yanks, where the 'really big shews were' to paraphrase Ed Sullivan.
I cannot tell you the numer of times that I have read newspaper or magazine reports of D-Day which refer solely to the British and American efforts. Small little unimportant Canada doesn't get a mention. I recall particularly an essay in a Norton Anthology of Essays about D-Day. A footnote commemorated the American and British efforts, without a word about the Canadian troups at Juno. (Let alone the various other battles such as at Caen which the Canucks won. Or the fact that, in World War I, it was a major Canadian offensive which began the last Hundred Days' battle which finally ended that war. Or Ypres.)
My point, other than personal jingoism? That history is written by the victors and their successors and when circumstances change, the successors may not necessarily bother with what really happened but might be more concerned with the alliances and allies and rapprochments of the current age in which they live.
Why did the annals of the Men make no mention of the Hobbits? Oh, I bet they were there alright. At the battle. But maybe memory of them faded as contact dwindled or their importance to Men lessened as Men regain their strength. Like the contributions of women throughout history, that of the hobbits became invisible. (oh, the irony here.
) And good empiricists simply shrug that the evidence is not as fullsome as they would like.