If they hold a trademark to the name, you cannot use it. Period. The only way around is to prove that you were first in use and that means you will need to challenge their trademark. While Apple isn't your problem, it is the cost of fighting this.
As the holder of multiple trademarks, I will tell you defending them or challenging a registration after it's been registered is a very expensive ordeal. I would try to work out a deal with them before I went to court. On the other hand if I were them and held the trademark, I would fight tooth and nail to retain my rights at all cost.
Topic:
App Store Distribution & Marketing
SubTopic:
General
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