For a browser story that talks to your own AI backend, HTML Game is the usual kind. Upload a small client (ZIP with index.html, or a single HTML file) so it embeds on the page, and have that client call your service over HTTPS. itch.io's HTML5 docs say any external API has to use HTTPS or the browser will block it: /docs/creators/html5
On the edit page, fill in Generative AI disclosure and tick Text & Dialog (plus any other types you use). The quality guidelines also ask you to state AI use clearly in the description: /docs/creators/quality-guidelines
I'd put the account requirement, usage limits, and that it needs a live connection in the first lines of the description, and again on a click-to-start screen before the story begins. Their guidelines also say to avoid third-party login walls unless they're actually necessary for the project, so keep any signup as light as you can.
Hope that helps Dorothy, figuring out how to label something that sits between a game and a web app is a fair question.