Ok I completely agree with Cherie - this horse HATES tongue pressure, get her a bit with a medium port or on that's well mullened. They make these in curbs AND snaffles, get whichever fits your riding style. Don't buy a rubber one, if she fusses with the bit that much she'll chew it up and the rubber makes sharp edges which hurt their tongue.
My next SERIOUS complaint is get the side reins off the 2 year old. She has a baby mouth, just learning what a bit is, side reins do NOT have enough give to have contact and release. She's a baby, she needs to learn the basics before you start messing with her headset and all that.
First off lunging two years old beyond a gentle walk and short trot is NO good for their joints, they're not nearly full grown yet and running on circles is stressful.
Next, she needs to learn how to give to gentle pressure before you apply something with as little give as side-reins. Personally, side reins make me furious, there are very few people who know how to use them right. I'm not trying to judge you're ability to use side reins - but THIS tells me it's not right:
"Once she has learned to seek relief one side at a time, I then put both side reins on, so she's straight and can turn her head in or out just a bit but for true relief she has to drop her nose vertical or a little in front of vertical. Since she wants to fight when you touch her mouth for ANY reason, I've tied her down pretty tight so that she is the one pulling on her face and she has to give herself the relief."
To teach a horse how to give to the bit here are the steps I follow:
Put a bit on and wait for them to be comfortable and settled. If they just can't settle in the bit find something with a different mouth piece style, of course give her time to get comfortable in it though before just switching it up. Remember adult horse's gums get used to the constant contact, but a 2 year has a soft baby mouth so constant contact hurts.
Once the horse is standing comfortably in their bit it's time to learn to give to pressure. I'm assuming you've taken the correct steps in teaching the horse to give to all sorts of pressure all over her body, yielding her hind and front end and backing up and putting her head down all with quiet cues? Next I'd start with a halter, flat not rope, and clip on reins to it. Hold on rein and rest your hand on the horse's wither, hold the rein so it's just taught, just a tiny amount of pressure and wait. Give her a minute, if she stands and does nothing, slowly apply more pressure (very slowly) if she starts fussing a little you have enough pressure, if she fusses big time you have way too much. So once she starts trying to figure out how to relieve the pressure, without freaking out that's the pressure just wait, when she finally turns in the correct direction relieve the pressure and tell her she's wonderful, even if she only moved a quarter of an inch in the correct direction. Gradually ask for more in each direction, this should take 3-4 15 minute sessions, maybe more or less depending on the quality of your timing. Eventually you will apply just gentle soft pressure on one rein and she'll bring her nose around to her side.
Now do the same thing with the bit she's comfortable with, remember that with the bit you'll need even less pressure.
What you're telling me in all your posts is that the horse is freaking out and she's bad - But honestly it just sounds like way too much pressure is being applied. Side reins are for adult horses who are learning how to work correctly, not babies who are just learning to give to pressure.
I'm not trying to be mean or be too pushy on you but I'm seeing a horse who's got way too much pressure applied and she's responding how most reactive horses would to that much pressure.
The other thing that concerns me is this quote "Ivy is somewhat pushy, but friendly enough on the ground, and is very responsive as long as she agrees with what you are doing."
She's 2 years old her attention span is still limited, keep your lessons short and sweet. But make them effective! If she's 2 years old and still pushy she's only going to get worse. You are her leader and her boss, she can't just do what you want when she wants to. Go back to groundwork basics. Yielding hind and front end, backing up and putting her head down. I mention putting her head down, despite it not being on most people's training schedules, but when a horse put's their nose down to the ground they shift their brain out of flight, reactive mode. Your horse is very reactive, all be it because there's something extreme to react to, but regardless teaching her to put her head down with a tiny ounce of poll pressure will really help you and her collect herself when she gets upset. We have a horribly flighty Arabian who we have taught this skill and now she does it for herself, when she starts to get worked up she throws her nose to the ground and dances in place for a minute until she's calm. She knows how to contain herself, we'll even see her do it alone in her field when something 'scary' happens. We've practically taught her a new version of 'flight', this is what relieves the pressure of whatever is frightening her, because when her head is down she's no longer frightened. Of course this is a band-aid to keep you both safe until she's properly desensitized and trusts you well enough to listen to you and not make her own decisions.
I'm sorry again for sounding so harsh. Please consider taking things slower with this horse and gentler, side reins don't have proper give or proper timing and they are too strong for a baby mouth. Please keep the bitted lessons short until her mouth is more comfortable with the bit, gradually start increasing the length of the lesson.