BillyS_ you should be able to re-use the API endpoint of the extension. You can take a look in the browser network tab under development tools to see the format of the request. If I remember correctly a POST request is made to /api/passwordless-request when Passwordless is enabled, and I think the payload is formatted similarly to the regular login payload from Flarum.
But hitting the API will just cause the email to be sent. You will probably want to modify the logic for your use case. Also my extension never validates password since the whole point is that email serves as the authentication factor.
You can see the source code for the controller here https://github.com/clarkwinkelmann/flarum-ext-passwordless/blob/master/src/Controllers/RequestTokenController.php
If you control the webview, you should be able to set cookies and you wouldn't need anything else than native Flarum API. You can directly create and set the remember cookie?