I know this post is dead but I was bored and reading old posts.
If you want the LEFT lead your ask with the RIGHT leg.
Actually, no. The inside leg at the girth cues the transition. The outside leg is passive slightly behind the girth, simply there to prevent the hindquarter from falling out.
You move your RIGHT LEG back behind the girth to push the hunches towards the inside of the ring, you pull the RIGHT rein to move the forehand towards the outside of the ring and cluck to the horse to break into the lope at the same time.
No. You don't move the haunches or the shoulders anywhere for the transition. If you start moving either, then you make the horse crooked and thus the transition more difficult. The horse should be 'straight' when asking for the transition and should 'remain straight' during the transition.
You don't pull on any rein.
Move the LEFT leg forward openning that side up to allow the horse to escape in that direction.
By doing this you are burying the right leg and exposing the left leaving little option for the pony to pick the right lead.
With a little practice the amount of crabbing you do to bury a leg/lead become less and less.
Inside leg stays at the girth and gives the aid for the up transition. The only thing you 'open' (or 'give') is the inside rein so the inside fore/inside shoulder is not blocked and thus the horse can jump forward into the lead once the outside hind begins the transition.
And this is why people have such a hard time teaching canter leads. They want to throw the horse off balance, make him crooked and all sorts of other things.
One must understand what's going on. The horse is going to have a weak lead because he's going to have a natural crookedness (moving with his haunches off to one side) because one of those hind legs that begins the depart just isn't going to want to bear the weight. The green horse is thus going to try and take the canter lead crooked and from the inside fore, rather than the outside hind and this is what gets them mixed up with their legs.
Before attempting to teach canter to a green horse, the horse should be relatively straight and supple and willing to bear equal weight on both hind legs.
In the beginning, any canter lead is a good canter lead, work with what the horse gives you and praise the try.
There are a number of exercises that can help set the horse up for the correct lead and if the horse is having troubles then you need to go back to the trot and work on gymnasticizing the horse and then lead into the canter with those exercises.
The thing I didn't like about what was going on here is that the OP was trying to make a deadline for a show. I am completely non-supportive of schedules and time frames and deadlines for horses. Puts too much pressure on horse and human and often times leads to steps being skipped, which ALWAYS catch up to you.
It's also quite normal for a horse to get the departs one day and not the next. They are not machines and neither are the people riding them.