Make captcha check over POST rather than GET

What does this MR do and why?

Make captcha check over POST rather than GET. See https://gitlab.com/gitlab-org/gitlab/-/issues/363154 for the reasoning behind this change. The POST endpoint was implemented in !88966 (merged).

Screenshots or screen recordings

No visual changes.

How to set up and validate locally

1. Setup Arkose Protect by following the instructions in https://docs.gitlab.com/ee/integration/arkose.html#configuration.
2. Sign out of your instance.
3. In the sign-in form, type your username in and blur the field.
   • An AJAX POST request should be triggered.

MR acceptance checklist

This checklist encourages us to confirm any changes have been analyzed to reduce risks in quality, performance, reliability, security, and maintainability.

• I have evaluated the MR acceptance checklist for this MR.

Related to #363154