Port published API data loading to the v13 website branch #1433

The compiler publishes versioned API snapshots to the website repository's default branch. PR #1432 makes website master consume the latest v12 publication, but the website's v13 branch also needs a loader that consumes published v13 data before compiler v13 CI and releases can target that branch.

Target branch: website v13.

• Port or adapt the published-data loader from fix: load API docs from published compiler data #1432 to the v13 branch's layout.
• Configure the documentation major as v13 and keep the displayed version, route generation, sidebar, page content, and API link validation consistent.
• Require a complete selected snapshot with valid library roots and module fields; fail invalid publications during the build.
• Consume the repository-root data/api/v13.* publications, with no reads from obsolete snapshot copies.
• Verify that API-data pushes to v13 trigger its website deployment; adapt the branch's deployment workflow if needed.

Acceptance: build the v13 website using a compiler-generated v13 snapshot; verify its root and module routes, documentation, and sidebar entries. Cover major/version selection and invalid publications, and run the applicable website CI checks.

Land this before switching compiler v13 publication to website branch v13.

Related: #1391, #1421, and #1432.

Coordinated compiler follow-up: rescript-lang/rescript #8753. That workflow change depends on this website preparation.