See #14 and #17 in the following thread: http://community.callofduty.com/message/414852149
14) Why am I being randomly moved from clan to clan?
This is a known issue that they are investigating. Here is a tweet from @Beachheadstudio on 11/7/2013. Twitter / beachheadstudio: We are aware of and investigating ...
17) The app/game are stating that I'm in two clans or the App says one clan and the game says another!
Some people's rosters are all screwed up due to this where in-game it shows 5 members but in the app it shows 4 or for others it shows someone in the clan but they are obviously glitched as when you go to Clan vs Clan it doesn't work. Attempts to leave one of the clans is also futile as you constantly get errors saying things like "You are no longer apart of this clan". Some are reporting that the app sees them in one clan but the game sees them in another (or not in one at all). I believe these all stem from the same data issue where they are somehow associated with 2 different clans.
This is a known bug/issue that may be related to #14. They are aware of this and looking into it, but no word on when a fix is expected. You can try to log out of the mobile app and log back in (forcing a refresh of your data). I have not heard of any workarounds to this either so your only option seems to be to have patience and hope Beachhead fixes this. You can contact them on Twitter @Beachheadstudio and see if they can give you any additional assistance but don't expect much.
Here's a possible solution to what you are trying to do:
See #17's 11/14 update: Clan Issues and Fixes
11/14 UPDATE: If the app says someone is in your clan but they really aren't, here's how to fix it:
1 - Have the user Log into the mobile app
2 - Go to the My Clan section
3 - Click on the Leave Clan Icon
4 - Log out of the mobile app
5 - Repeat items 1-4 until you are no longer in a clan (they may find themselves in a few different clans)
6 - Have the person invite you into the clan you want to join.