I would love to say Dead Rising, but after 10 years, it recently got re-released on PC, X1 and PS4. Not sure it counts anymore, it did back in its day (it's not a remaster per se, it's just a re-release of the same game for these platforms, which I'm not sure it can be included on the reasoning OP gave).

There's Fable 2, which is an amazing game, full of personality (though very easy and a tad streamlined for RPG standards, even more if you compare it to its predecessor). Definitively a must if you owned a 360.

Lost Odyssey is magnificent, despite some issues it had (like being the slowest-paced JRPG ever conceived). 360 is the only option if you want to play Tales of Vesperia and you don't understand japanese or don't own a not-updated pirated PS3 and are willing to import (and even then, the extra ps3 content is still on japanese, so there's that), so I would like to include that one, but on a less "exclusive" degree than the rest given its circumstances.

Beautiful Katamari was charming, despite Bamco doing the worst microtransaction bussiness out of it I've ever seen.

All in all, my pick is the original Dead Rising, followed by Lost Odyssey if the former doesn't count, and Shadow Complex in the third place.

I'm disappointed. You haven't mentioned Akai Katana so far, which was vastly superior to both Doddonpachi and Deathsmiles. >:(

Isn`t that disqualified due to its later release on PS3 though?