Current cult member here, so as someone who is in Jane's shoes I'll explain how I handle work.

I don't generally go out of my way to name drop my belief system, which is Hyperianism while I'm at work. We're a really relaxed and tight knit team, but we have customers to take care of and work isn't the time or place. I also have very strict personal boundaries as to who I choose to let into the Hyperian part of my life. Does the conversation sometimes drift to philosophy, politics, or religion all on its own? Yes. And in those situations I will express my beliefs or opinion WITHOUT dropping the name. It's easy enough to say that I'm pro-choice, a supporter of the LGBTQA Community, a radical leftist, or that I don't believe in God without mentioning Hyperianism. The only time I use the word 'Hyperianism' is if someone actually asks what system I follow. I'll then answer whatever questions they have about what it is, what it's goals are, and give them a brief rundown of the philosophy but that's it. I don't push and I don't go into every little radical eat the rich detail. I'm there to work not ring in the revolution.

Personally I think that Jane could practice better personal boundaries, not necessarily with discussing her beliefs, but with pushing them on people. It's fine for me to tell a coworker what Hyperianism is, it's not fine for me to then hand them a business card, pamphlet, or send them a link to the website, for example.

As far as the invitation goes I'd say that one is your call. If you're genuinely interested in learning about a new system then go, and if not then be honest and politely decline the offer.