I'm not a teacher, so I have no real knowledge of what to teach studentsof English. However, if you teach standard English, are there really that many differences between British and American varieties when talking about grammar?
Most differences lie in idioms and expressions used, slang/informal vocabulary, and of course spelling. Even so, they are often similar.
I would say spelling requires you to choose whichever variety English you are comfortable teaching consistently. Spelling some words using American, and then other using British would be confusing.
It will be interesting to see what actual teachers think from their experience. Especially non-native teachers.
Student or Learner