Incorporate Keycloak login into SPA
Even if it's not considered as a best practice you can turn on Direct Grant Access for your client which enables to login through a REST call.
Anyway, about not loosing the state of your app, this is a bit outside the scope of Keycloak but you should be able to achieve that with having the state in your redirect URL for instance ?
Also, if you don't want your app to automatically reidrects to the login page you can use : keycloak.init({ onLoad: 'check-sso' })
instead of login-required
For someone who comes back to this question,
I think it's better to stick to the best practice for oAuth2/OpenId Connect for SPAs which is currently "Authorization Code Flow" with PKCE.
https://oauth.net/2/pkce/
https://datatracker.ietf.org/doc/html/draft-ietf-oauth-security-topics-13
A normal flow here needs a complete redirect to the auth server and back so your app will completely re-initialize. Or you use check-sso like Sébastien already mentioned with silent mode.
https://github.com/keycloak/keycloak-documentation/blob/master/securing_apps/topics/oidc/javascript-adapter.adoc
You can configure a silent check-sso option. With this feature enabled, your browser won’t do a full redirect to the {project_name} server and back to your application, but this action will be performed in a hidden iframe, so your application resources only need to be loaded and parsed once by the browser when the app is initialized and not again after the redirect back from {project_name} to your app. This is particularly useful in case of SPAs (Single Page Applications).
This way the login will happen in an iframe and the app initializes only once and should preserve state.