Authentication with fingerprint in flutter webview

Question

I'm learning flutter webview. I want to authenticate with my fingerprint. The site that I wrap in webview is written in Laravel. I use local_auth 1.1.11 as an authorization plugin. How can I combine this plugin and authorization on a site that is wrapped in a webview. At the moment, I'm thinking of setting up the application on the side of the flutter itself (the checkbox, when enabled, will enable authentication using a fingerprint), but where to go next? If I make this setting, how can I authorize on the webview site when scanning a fingerprint? As I understand it, it would be a bad idea to manually enter the login password in the same settings on the flutter side and then send this data to the site as inputs when scanning the fingerprint? Please help and tell me in which direction to move and how it can be implemented.