Actually it does works fine with even Xcode 7.
One possible reason might be, for using <TeamchatSDK/TeamChat.h> instead of <TeamchatSDK/Teamchat.h> (Theres a slight mismatch with class name and file name with 'C' in Teamchat.h).
while specifically importing selective class.
If that doesn't help, Can you please confirm if the steps you followed is same as in below youtube link? if issue is specific about only SDK Integration.
That might help fix the issue.