feat(nextjs): Inject manifest into client for webpack builds

[chargome]

ref #16683

This PR injects the manifest into client bundles for webpack.

[github-actions]

size-limit report 📦

Path	Size	% Change	Change
@sentry/browser	23.88 kB	-	-
@sentry/browser - with treeshaking flags	22.35 kB	-	-
@sentry/browser (incl. Tracing)	39.75 kB	-	-
@sentry/browser (incl. Tracing, Replay)	77.88 kB	-	-
@sentry/browser (incl. Tracing, Replay) - with treeshaking flags	67.7 kB	-	-
@sentry/browser (incl. Tracing, Replay with Canvas)	82.57 kB	-	-
@sentry/browser (incl. Tracing, Replay, Feedback)	94.68 kB	-	-
@sentry/browser (incl. Feedback)	40.56 kB	-	-
@sentry/browser (incl. sendFeedback)	28.56 kB	-	-
@sentry/browser (incl. FeedbackAsync)	33.46 kB	-	-
@sentry/react	25.61 kB	-	-
@sentry/react (incl. Tracing)	41.7 kB	-	-
@sentry/vue	28.31 kB	-	-
@sentry/vue (incl. Tracing)	41.54 kB	-	-
@sentry/svelte	23.9 kB	-	-
CDN Bundle	25.16 kB	-	-
CDN Bundle (incl. Tracing)	39.41 kB	-	-
CDN Bundle (incl. Tracing, Replay)	75.38 kB	-	-
CDN Bundle (incl. Tracing, Replay, Feedback)	80.87 kB	-	-
CDN Bundle - uncompressed	73.5 kB	-	-
CDN Bundle (incl. Tracing) - uncompressed	116.89 kB	-	-
CDN Bundle (incl. Tracing, Replay) - uncompressed	231.06 kB	-	-
CDN Bundle (incl. Tracing, Replay, Feedback) - uncompressed	243.88 kB	-	-
@sentry/nextjs (client)	43.34 kB	-	-
@sentry/sveltekit (client)	40.2 kB	-	-
@sentry/node	167.58 kB	+0.01%	+1 B 🔺
@sentry/node - without tracing	100.3 kB	-	-
@sentry/aws-serverless	128.41 kB	-0.01%	-1 B 🔽

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