Jennifer -
If I may add in my 2 cents as someone who has dealt with dogs that have reacted very strongly to correction with a prong or any other collar other than a DD...
One thing is definitely to be watching body language. Keep your eyes on her ears. You mention she is very alert and that her body language seems to be confrontational in how she presents herself to other dogs as dominant. Some dogs can react very strongly to a correction, no matter if they have already gone into aggressive mode or not. Some dogs become aggressive towards you for a correction. This tends to be a pack issue/dominance problem in most dogs, the dog is correcting you for correcting it. The action of the prong is like teeth on the dogs neck, that is one of the reasons dogs respond so well to it, and also why dogs can respond poorly to it. It goes something like this: Teeth on neck - challenge/attack on them - fight back - bite you.
Work on Teagan with focusing on you during the walk, not on her surroundings. You as the alpha should be the only one displaying dominant body language. As Carol suggested work on the "look" or "see" or "face" command (to get the dog to look you in the face) Some dogs it isn't so much as looking to you for guidance it becomes ingrained in them that when they see another dog, they get praised/rewarded for looking at you, you create an association of: see dog - look at mom - get reward!
Change things up a bit to make her pay attention to you and what you are up to, not what every living thing around her could be doing. If you notice she is looking at everything, sniffing the air, ears forward, do a sharp turn, bring her attention to you. Ask for a sit, a down, a shake/paw, a front, anything. Go back to the walk, as soon as her attention leaves you, another turn, or a stop, or an about face, or she is being told to sit or down, etc. The walk is about exercise, yes, but about communication and about YOU, not Teagan, It isn't Teagan's walk. It's your walk that Teagan is being allowed to accompany you on.
Easier said than done in a lot of cases, but Teagan sounds like she is generally a good dog. You can try taking a favorite toy with you to help her focus and distract/redirect her if she seems to be flipping into a prey response towards a small dog or animal. I would take Teagan out alone, without Luc, to be better able to completely focus on her. Look at the walk as more of a training session than a walk that could be interupted by a problem.
Also.. know that any feeling you have is transmitting to her no matter if you are holding the leash or touching her or not. If you are nervous, she will know this, and will respond accordingly. Give her the benefit of the doubt even if you do not trust yourself and worry about your timing. Remaining calm and focusing on her instead of the dogs you could possibly encounter, will prevent Teagan from feeding off of what you are feeling in a negative way.
For prong corrections on a dog that becomes aggressive towards me, or is escalating because of the correction, I find that moving backwards and correcting back and up enough to cause the dog to loose balance can help in diffusing the situation. You are not moving away from them, you are bringing them back with you, away from the problem. Moving back allows for a stronger correction but with a prong you do not want to lift the dog as you would with the DD, it would be for purposes of causing the dog to lose balance only.
Unless all you have is the prong and the dog continues to escalate and is becoming very aggressive and is trying to, or succeeding in biting you, then I would indeed hang the dog with the prong as a last resort to keep from getting torn up.
I would suggest a DD since she has already escalated. Timing is important but with aggression, a dog that escalates and just becomes more aggressive with a correction, for safety issues, should be moved to a DD. If you correct her while she is showing aggression, she is not always/necessarily more likely to escalate than if she had only just noticed the other dog. If she is not already beign aggressive, she could become defensive towards you for your correction, as her attention was on the "intruding" animal, you corrected her, either catching her offguard, or causing her to become aggressive towards you (as her auto-response to correction/an aggressive action towards her). Not saying this WOULD happen, but it depends on the dog, the situation, etc.
A dog that just moves up in aggression level once, is more likely to do it again, than a dog that never has, no matter the timing. You are either causing her to become aggressive to begin with, or making her more aggressive than she already was. It could be an auto response to become aggressive about a correction - you want to stop the automated part, you want to break the cycle.
I also agree with using more than one collar, I would start off with the prong and the DD. I actually use 2 leashes when I am introducing the prong to young dogs. I have a traffic lead, 12" is usually plenty more than enough for a larger dog, shorter is better the taller the dog is. Leave your regular lead on the prong, traffic lead on the DD. Adjust the prong to right directly behind the DD, this shouldn't be an issue if it is already correctly fitted, as the DD is not a very large/wide item. I know some may argue about this because they are both supposed to be in the same spot, but the prong is still effective this way, I have used this method.
Ok, so it was more like 5 cents