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:
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?