Migrate from Better Stack to Sentry (Cloudflare Workers module) for error logging

[baronunread]

Context

Today alertBetterStack() in src/worker/alerts.ts is the only external logging/alerting integration. It's called from just two dead-letter-queue consumers (clicks.ts, storage.ts). Everything else (global error handler in index.ts, domain CF-API failures, rate-limit warnings, batch-insert failures) only goes to console.error/console.warn, i.e. Cloudflare's default log stream, with no alerting or aggregation.

Goal

Replace Better Stack with Sentry using the official @sentry/cloudflare Workers module, and use the migration as a chance to widen coverage so errors that are currently only console.errord get captured too.

Proposed work

• Add @sentry/cloudflare, wrap the Worker fetch handler with withSentry() (per Cloudflare's supported integration), add SENTRY_DSN (and org/project ids if needed) to Env, wrangler.jsonc vars, .dev.vars.example, and prod.secrets.env.example.
• Replace alertBetterStack() call sites (clicks.ts, storage.ts dead-letter consumers) with Sentry.captureException/captureMessage.
• Wire the global error handler (index.ts:46) and other console.error call sites (domains CF-API failures, rate-limit warnings, batch-insert failures) into Sentry so they're actually visible/alertable instead of silent log-stream noise.
• Remove alerts.ts, BETTERSTACK_SOURCE_TOKEN/BETTERSTACK_INGEST_URL from Env, wrangler.jsonc, and secrets docs; keep the playwright env's no-op-in-tests behavior equivalent for Sentry (disable/no-op in e2e).
• Update AGENTS.md/CLAUDE.md Config section to reference Sentry instead of Better Stack.

Out of scope

Source maps / release tracking can be a fast-follow, not required for v1.