There's a few solutions to the problem. Hopefully one of these will work for you:
1) You can add the blue Facebook login button into your CB module. To add the button, you'll need to add a little bit of code to the module, as explained:
www.sourcecoast.com/extensions/jfbconnec...estions#login-button
If you do this, since the CB module's logout button will not log a user out of Facebook, you should not enable the "Autologin Facebook Users" option.
2) You can use both the JFBClogin module and CBLogin in different positions. Edit one module to hide the logout button and the other should work as normal. (You may want to add the Login With Facebook button to the CB module as mentioned above)
3) There's no 'built-in' way in Joomla to use multiple default.php files depending on the instance. However, you can do this pretty easily on your own. Simply copy the mod_jfbclogin directory to mod_jfbclogin_special. Then, edit the mod_jfbclogin.xml file and update the name to "JFBCLogin Special". You can modify this special module's default.php however you want, and when you're in the module manager, you can add it just like a separate module.
Hope one of these solutions help. If you have questions or problems, just let us know!