It can be done as a Summoner *maybe*, but you have to abuse the way the aggro works on the mobs.
When you assault a mob with your summon (there are 3 in the room) the other 2 will auto link to your avatar. When your avatar dies, as long as you did not perform any enmity actions on the other two, they will not aggro on you when your avatar dies, because they linked. You basically abuse this and take them down one at a time.
Some of them are rough, and they have their own mechanics; you would need to be geared, not only for survival, but also to be able to kill them fast enough to meet the time limit. I can't confirm either or not it would be doable, especially if you don't have Nirvana, at the very least due to the time limit.
You can see Papesse soloing the Hall of An here, for an idea of how to do it (they have Nirvana):
https://www.youtube.com/watch?v=TGGFvtnTidY
...but either way, it would be better to do it in a group. You can kill the Veiled Ironclad 4-6 times in a 6+ man group within 1 run, and each run costs a decent sum of gil. With such a low drop rate on the earring, it would be much more efficient in a group.