I'm not a native Brit, so here's just my idea for an alternative expression:
mechanisms by which national identities are constructed as an opposition to a foreign Other.

to play off of sth. does not mean what #1 suggests, so to me, that's wrong. However, I'm not a BE speaker, so I can't answer your question about BE usage.

I do not understand "play off of" (as used in #0 at any rate). AFAIK, "off of" is not standard UK English, though my impression is that it's used regionally, i.e. in one or more UK dialects (e.g. West Country?).

It's my (strong) belief that "off of" is wrong in AE, too. But, unfortunately, that does not stop a lot of Americans from using it anyway.

The suggested alternative is not bad, but I'm not completely convinced.

The other suggestions are not particularly helpful. Here's the reference I found yesterday:
2
play off somebody/something
American Englishto deliberately use a fact, action, idea etc in order to make what you are doing better or to get an advantage:
[Look up a word starting with D or S for samples of headword or sentence pronunciations on the LDOCE CD-ROM] The two musiciansplayed off each otherin a piece of inspired improvisation. (http://www.ldoceonline.com/dictionary/play_1)

It's true the the "of" is not used here. Is it consensus that it is wrong? (Is there another US-speaker to back me up here?) Should I just say "...mechanisms by which national identities are constructed which play off representations of a foreign Other"? Is that understandable to British speakers?

I'd say it's primarily an American colloquialism, much like "get off of me" is, or "take this off of me." It may be wrong to the purist, but it's so widely used, it might as well be correct for AE.

http://www.collinsdictionary.com/dictionary/e...
A lot of things go through Vince (Carter) and you just have to play off of that.
TORONTO SUN (2003)