feat(web): add Better Auth client integration (#114)

[RAHULDINDIGALA-32]

Summary

Integrates the Better Auth React client into the web application for cookie-based session authentication with the Express API. This PR also fixes a workspace TypeScript export-resolution issue that prevented the Next.js app from building.

Changes Made

• Added better-auth as a dependency of apps/web.
• Added apps/web/lib/auth-client.ts.
• Configured the auth client with NEXT_PUBLIC_API_URL.
• Configured credentials: "include" for cookie-based session handling.
• Exported authClient, signIn, signOut, useSession, and getSession.
• Corrected @repo/types barrel exports to use extensionless source imports, resolving Turbopack’s inability to resolve non-existent .js files from TypeScript source.
• Updated bun.lock.

Testing

• bun run lint passes.
• bun run test passes: 11 tests.
• bunx tsc --noEmit passes.
• bun run build passes.
• Manual testing completed
• No regressions introduced

Related Issues

• Fixes [Task] Better Auth Client Integration in Web #114.
• Includes a required build-blocker fix in @repo/types, necessary for the Next.js application to resolve shared form-schema exports.

Screenshots (if applicable)

Not applicable, as this PR adds client-side authentication infrastructure without visual UI changes.

Checklist

• Code follows project conventions.
• Tests pass.
• Documentation updated where needed through this PR description.
• No linting errors.

[coderabbitai]

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[Basharkhan7776]

Merging this PR, all clear