[TalnSG]

Take the safer route and stick to at least a western accent. As a former drame teacher I think I understand the intent of the exercise, so perhaps these would suffice.

Strong southern drawl
Western US "cowboy" with strong "twag"
Brooklyn (very distinct and frequently used)
German
Swedish

I would have included Spanish or Italian, but unless your are reaaly good its going to blur with French for most listeners.

And since you mentioned English is too common, are you referring to the Queen's English, Cockney, Welsh or Scottish? They vary considerably when done correctly.

To do an accent that you cannot back up visually on stage is a wasted effort, since you will only be cast if you physically fit the role as well as vocally. You mentioned being part Asian, but the issue is the visual image nor the genetics.