Can I switch the site being used as provider (staging to prod)?

Yes, you’re right, it is a relatively common scenario.

Nevertheless, if your Wordpress is in production, as seems to be the case with this chap (@Firsh please correct me if I’m wrong), then I think the potential risk / cost of mixing staging and production environments is outweighed by any time and cost saved by just spinning up a separate Discourse instance. This is assuming your staging is a copy of your production (which is often the case).

Unless there are organisational restrictions on spinning up new instances (in which case you should probably have a separate staging environment anyway), it’s relatively cheap and easy to spin up a separate Discourse on DigitalOcean, Vultr etc, and avoid any issues that might arise from Staging <> Production cross-pollination. You’ll probably save time (and money-as-time) in doing so. You can just turn it off when you’re done.