first and foremost, showing in a few weeks imo is a bad idea and the horse isn't ready for it. You mention a few things that make me want to go back to the basics with this horse:
this mare is very on her forehand, and doesn't like to bend or use herself at all
horse is not lame, but she picks up correct leads in round pen
this tells me the horse is having an issue balancing with a rider, and that they are doing the flying change the way they were taught on the track (switching at full speed with a rider over the withers is VERY different of a maneuver than switching at a balanced working canter with a rider on/over the back).
Forget everything else, forget HOW you are asking her to canter on the right lead, and go back to trot work. Start with trotting on a long rein so you are not holding the horse up. Forward first, frame later. Get her moving forward so she eventually has to pick herself up. Ignore where her head is, ignore whether she is on the fore or not for now, and focus on her moving forward with impulsion. As she builds muscle in her topline and hind end, eventually she will begin to work off her hind rather than leaning on the fore. As this starts to happen, add changes of direction. Figure 8's, serpentines, random weaving and figures to really work on the horse suppling through the spine. A horse on the fore will not leg yield or shoulder in properly and will simply just be counter-bending to complete the maneuver, thus reinforcing the WRONG muscling which is why you aren't seeing a difference in the ability to get the right lead. The horse isn't ACTUALLY suppling.
By doing various figures and movements, the horse will start to have to balance themselves. Again the more you can do on a long rein, the better. When you try to hold the horse up and/or use the reins to steer, that opens the door for false framing, popped shoulders, dropped shoulders, crooked hips, leaning on the bit, and balancing off the rider. The horse needs to move off the leg and through their hips and back.
To test your progress, start frequently changing diagonal without changing direction every few strides. When you can change diagonal on the straight AND the bend intermittently regardless of what direction you are tracking in, AND the horse's body, head, and balance are unaffected, that is when you know the horse is balanced throughout and moving properly. If the horse pops up their head, drops a hip or shoulder, or leans when you change diagonal, that indicates there is a weakness and go back to more trot work.
When the horse is able to balance through the trot, then you can add in transitions to further engage the hind. I have a 3 stride rule - if your horse can't come to the requested gait within three strides, then the horse is not truly balanced and "on the aids". If you are using your reins to steer rather than as an accessory to reinforce the request of the leg, the 3 stride rule will bring out that fault rather quickly.
Once the horse is properly and evenly muscled, the canter will be much easier, regardless of lead. At that point, when ready, the horse should be able to canter by bending to the inside, adding the outside leg to encourage the horse to pick up the inside shoulder (and prevent the horse from diving in or bracing/leaning on the inside aids) and be able to move into the canter with a light rein contact off the hind.
In addition, all the above will make for a much better muscled horse, smoother more balanced ride, and better placings at ANY show, regardless of discipline because the quality of the movement will be dramatically improved.