kjartan - At present, Lithium doesn't provide much of a control over customizing this page as we only have one encapsulated component on this page. So you need to contact the support and involve Professional Services to make a move further. They add similar fields to the user profile for us and they are available in the APIs as well the user metrics report, so you can confirm from them if the added fields be available in the REST calls. Once they add the custom fields, you can do whatever you want.
I wouldn't recommend doing it via JS on this page, this is a dynamic page which renders differently for other content types, and permissions, though using JS won't be an issue but you'd never want to risk anything on this page which could potentially keep off your user to post/ reply on the community in case of slightest of errors. That's why I'd recommend you to ask PS if they can add a field for you on this page and provide a possible API for this.
Let me know if this helps.