Persistant patting of the service dog is a dick move and not acceptable.
However, public self-description is an example of where attempts at inclusivity/reasonable adjustments challenge problematic attitudes and reactions can cause problems.
Both the request and the discomfort expressed can be reasonable responses and also both true, at the same time. Asking for a description of yourself seems perfectly reasonable to accomodate someone who can see what you look like. But not everyone knows how to respond (or even why it's useful) and some people can't respond without it raising issues for them. All of this depends on how the question was phrased; it was a training event after all, and therefore explaining needs to be part of the process - eg what does that person want to know, what you can usefully or helpfully say, what you might not want to say, how much to say, in order that people can work out something that is both accomodating and helpful, and comfortable for all parties). Of course if all this was done, and Person B reacted the same way, they are being a PITA and clearly don't want to get it.
It is a new thing for a lot of people, and the whole point of training is to address ignorance, inform and improve skills. The attitude that "everyone should know this" is naive. While personally I think Person B is both a dick and holds some probably quite ill-considered views, aren't they exactly the person the training is aimed at? Helping them see how to behave in a non-discrimintory way, possibly to find a middle-way through their issue about self-description (which maybe not about bloody-mindedness or predjudice, but about fear and anxiety). Disability awareness training isn't about already knowing what is best thing to do or there wouldn't be much point and you wouldn't have called anyone in to do it.
I think it is really hard to do any kind of EDI training and especially if you have lived experience of the protected characteristics too, and a trainer should have the support, strategies and protection of their own, and that of the company if they feel threatened or discriminated against. Did you not get a set of ground rules and expectations for the session?
I have seen trainers leave a session for a break, opt out of session until the work until the group addresses some of the issues arising themselves, refuse to work with cetain companies and all of these are reasonable self-protections and can actually be a very effective tool in training. It's exhausting and not very rewarding work. But at the same time the point is to deal with the problematic attitudes as they arise. If the trainer felt Person B's behaviour was beyond their acceptable boundaries, then they need to be listened to, but perhaps they might be able to suggest how this might be dealt with? if Person B is clearly not open to the training, that needs addressing - but at a "this is what this company believes and this what is expected in your role" level.
I also think that EDI training should be required (for everyone including all senior and executive staff) and should be part of corporate strategy. It isn't a bolt-on or optional if you feel like it, or a sticking plaster. I am slightly concerned that neither you nor the head of HR were at the event to mediate/observe. Had steps been taken to look at risks for trainer and staff or take the temperature (clearly cool to luke warm by the up take numbers) or to thoroughly investigate what the trainer was going to do or say and if they were a good fit for where your company is actually at, at this point. Training resulting in a complaint of this kind suggests that none of this was worked out before the session took place and that a lot more work is needed by the company and it's unfair to any trainer you employ, until this is done.
I feel like both Person A and B have been set up to fail here - not necessarily your fault - but working in a company that isn't actually interesting or embracing this work is a hiding to nothing.