apply customization for dynamic site context in v2

[conico974]

We never applied the customization settings in V2. This PR fixes it

[linear]

RND-7290 V2 Preview (Vercel) is not working

[changeset-bot]

🦋 Changeset detected

Latest commit: f585ce7

The changes in this PR will be included in the next version bump.

This PR includes changesets to release 1 package

Name	Type
gitbook-v2	Patch

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Click here if you're a maintainer who wants to add another changeset to this PR

[github-actions]

Summary of the deployments:

Version 1

Version	URL	Status
Latest commit	https://c795975f.gitbook-open.pages.dev	✅
PR	https://pr3302.gitbook-open.pages.dev	✅

Version 2

Version	URL	Status
Vercel	https://gitbook-v2-3nwg2mo1i-gitbook.vercel.app	✅
Cloudflare	https://edbcdcae-gitbook-open-v2-preview.gitbook.workers.dev	✅

Test content

Site	v1	2v	2c
GitBook	https://c795975f.gitbook-open.pages.dev/gitbook.com/docs	https://gitbook-v2-3nwg2mo1i-gitbook.vercel.app/url/gitbook.com/docs	https://edbcdcae-gitbook-open-v2-preview.gitbook.workers.dev/url/gitbook.com/docs
E2E	https://c795975f.gitbook-open.pages.dev/gitbook.gitbook.io/test-gitbook-open	https://gitbook-v2-3nwg2mo1i-gitbook.vercel.app/url/gitbook.gitbook.io/test-gitbook-open	https://edbcdcae-gitbook-open-v2-preview.gitbook.workers.dev/url/gitbook.gitbook.io/test-gitbook-open

[argos-ci]

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Build	Status	Details	Updated (UTC)
customers-v1 (Inspect)	✅ No changes detected	-	Jun 11, 2025, 11:46 AM
customers-v2 (Inspect)	👍 Changes approved	77 changed	Jun 11, 2025, 11:46 AM
default (Inspect)	✅ No changes detected	-	Jun 11, 2025, 11:51 AM
v2-cloudflare (Inspect)	👍 Changes approved	404 changed	Jun 11, 2025, 11:52 AM
v2-vercel (Inspect)	👍 Changes approved	404 changed	Jun 11, 2025, 11:54 AM