Despite Achilles and Ajax being first cousins, they weren't depicted as being chummy because Ajax was in the battlefield throughout 3/4 of the book and Achilles was sulking for that same period. Though Achilles does consider him and Ulysses friends when they do come to get Brieses, but there is no obvious "best buddy" closeness between them "IN THE ILIAD" due to the fact I mentioned previously. So therefore we cannot infer that Ajax should have been given armor due to being either his cousin or the closest to him as kin or friend.

And even though the armour was too big for Odysseus to wear, my understanding of the armour feud, wasn't which armour fit who the best, it was who was going to have the honour of receiving the greatest armour created for the greatest warrior. It was the recognition and the adulation that went with it, that they were fighting for. These Ancients had big egos, and one could not bear having another warrior claimed to be better for their cause, especially when one thinks they are greater (ex: Ajax). Besides armour size really doesn't matter. Patroclus seem to fit into Achilles old armor alright, Hector did too with help of Zeus, therefore with a God's help Ulysses could have fit in, though he was considerably shorter.

And even Achilles' own son did not dispute the right of Ajax to have that armour; mainly because Phyrrus did not get to Troy until after the death of Ajax. He came along during the "Euryplus son of Telephus was leading the Trojans" phase of the war. Ajax was already dead and buried by then. Actually to Ulysses' credit he did give Phyrrus his father's armor, when he fetched him on Scykros.