chore: pin prettier to 3.9.5 and run format check after install (#1716)

The "Check formatting" step in main.yml ran `npx prettier --check .`
before `npm install`, so npx fetched the LATEST prettier rather than the
version pinned in package-lock.json. This made the format check
non-deterministic and red on open PRs whenever the latest published
prettier differed from the pinned one.

- Pin prettier to exact 3.9.5 in devDependencies and update the lockfile
- Reformat the repo with the pinned prettier (expected source churn)
- Move the format check after `npm install --no-package-lock` and run it
  via `npm run prettier-check` so CI uses the locally-installed pinned
  prettier


Claude-Session: https://claude.ai/code/session_01HiccCEh9mwCfVzE8qYcop1

Co-authored-by: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

package-lock.json

@@ -38,7 +38,7 @@
         "jest-fixed-jsdom": "^0.0.9",
         "lint-staged": "^16.1.5",
         "playwright": "^1.56.1",
-        "prettier": "^3.7.1",
+        "prettier": "3.9.5",
         "rimraf": "^6.0.1",
         "typescript": "^5.4.2"
       },
@@ -10908,9 +10908,9 @@
       }
     },
     "node_modules/prettier": {
-      "version": "3.7.4",
-      "resolved": "https://registry.npmjs.org/prettier/-/prettier-3.7.4.tgz",
-      "integrity": "sha512-v6UNi1+3hSlVvv8fSaoUbggEM5VErKmmpGA7Pl3HF8V6uKY7rvClBOJlH6yNwQtfTueNkGVpOv/mtWL9L4bgRA==",
+      "version": "3.9.5",
+      "resolved": "https://registry.npmjs.org/prettier/-/prettier-3.9.5.tgz",
+      "integrity": "sha512-/FVl766LpUfB5vXgCYOYa0MeV/441Ia99AeICQIQFTY/Nw0roZwULcXpku5i1/m5kt/baz+s4Zogspd839HSMg==",
       "dev": true,
       "license": "MIT",
       "bin": {