Yes, like Izabella I wouldn't buy a clarinet on eBay unless it was local, I knew what I was doing and I could give it a good run-through first. Also, when buying second hand, have a good look at the case because rough-handling may be more obvious there.
For a beginner I'd recommend a modern, resin clarinet. They're lighter than wood and it is no longer true that any wood clarinet is better than any resin one.
If the player is of grade three standard or above, then she might find that she prefers one model over another, i.e. similarly priced Buffet or Yamaha. A player of grade five standard or above would benefit from an 'intermediate' rather than a 'beginner' model - funds permitting.
It is a Bb (B-flat) clarinet that you're looking for. Dawkes is a respected supplier; if it isn't on their list it probably isn't worth having. Windcraft is their own make, made to a budget so unlikely to be as good as the Buffet but probably better than anything else at less than £389. But I'm not up-to-date on it, so I don't know.
Dawkes clarinets
Definitely talk to the teacher.