I have modified the available Toto-Rak map to make it more informational and to show the first Imperial Devices Quest Moogle NPCs, which was missing from the image. Also, I added a better way to unlock the first Boss chest. No need to kill any mites in the dungeon at all. I have fully tested it on all three bosses and it has never failed. I farm seals there frequently and since I figured out which mobs to kill, the last 50~ runs in a row, each dungeon clear always has the 5 chests and I never kill a single mite (with the exception of the bosses. lol)

The way I figure it, there are probably 2-3 mobs in the dungeon which trigger the first boss chest, 1 in first area if you are only going after the first boss and either 1 more per additional boss area or the second & third have just 1 mob total (since you can't access the third without the second, but you can skip the first altogether.) First area mob is Bombs, Mites are 2nd/3rd, but all you need is one of them to trigger the chest, not both (or all three.) The Bombs one is right next to a Photocell and it's just 2 Bombs, where the mites ones has a bunch of mites. I find the Bombs route faster, anyway, hope this helps.

http://s17.postimage.org/9zxc9u1rh/Toto_Rak_Ox_Modified.png