BuddyPress.org

I would copy the entire bp_get_nav function except the final part where it adds the logout link, and make your own function called bp_get_nav_no_logout, and put that function in your bp-custom.php file.

This way you can modify the member theme instead of the core files.

Take this a step further and make your own member theme, that way you’re not modifying the stock member theme either.

If you feel like this is something you’d like to have more control over in the future, add it to the trac as a feature enhancement. I could see being able to hook into those functions or modify their behavior as having its benefits.