It’s fairly straightforward to set up, since Discourse provides the JSON and you can just read that client-side. I’m sure you could do it a more complicated way, but this is essentially how I’ve done it with Javascript. I modified my code slightly since you’re looking for email and name, but you should be able to just embed the below code under Customize > CSS/HTML and be up & running.
Note the two places in the code of YOUR_INTERCOM_APP_ID and YOUR_DISCOURSE_DOMAIN.
Also FYI the current Intercom Messenger doesn’t play well on iOS devices on Discourse. But they could be working on an updated Messenger. And if I were to have used an updated Intercom Messenger, I would say that it works very well on iOS now
I’ll run a quick test with this code to see what the issue is. Before I do, I assume that your Wordpress/Discourse user emails match up with users you’ve already loaded into the Intercom admin dashboard?
SSO isn’t an issue. I actually have the same set up.