feat(server): opt-in uncaught exception capture

[cat-ph]

💡 Motivation and Context

Third PR in the 4-PR JVM error-tracking stack: opt-in capture of uncaught JVM exceptions for the server SDK.

• New PostHogConfig.captureUncaughtExceptions (also on the Builder), default off. When enabled, PostHog.setup installs a global Thread.defaultUncaughtExceptionHandler that captures the crashing exception as a fatal, unhandled $exception event (mechanism UncaughtExceptionHandler), flushes, then delegates to the previously registered handler. close() removes it again.
• Unlike the Android SDK, this is gated purely on the local flag: the server SDK never fetches remote config, so the remote autocapture toggle can never fire. To support that, the shared PostHogErrorTrackingAutoCaptureIntegration takes an optional caller-supplied enabledGate instead of always using the built-in gate (local errorTrackingConfig.autoCapture + remote config as a kill-switch, i.e. the behavior from Bug: error tracking autocapture misses crashes on first launch before remote config resolves #648 is preserved for Android).
• The handler now delivers captures through an internal CaptureTarget seam (installWith), because the core PostHogInterface and the stateless server client share no capture supertype.
• With no previous default handler present, the handler reproduces the JVM's own Exception in thread "..." stderr output, so opting into capture never hides a crash from stderr log collection.
• New internal PostHogCapturedThrowables guard: a weak, identity-keyed, process-wide set letting independent capture paths avoid double-reporting the same Throwable instance. It is deliberately directional — the uncaught handler only marks (a crash is always captured as the authoritative fatal/unhandled record, even if the same instance was logged first), while log-mirror paths (the appender in PR 4) consult it and skip instances already reported. It never keeps a throwable or its stack alive.

Delivery caveat documented in the KDoc and changeset: flush() drains the queue synchronously on the crashing thread, but the preceding capture enqueues asynchronously, so under an immediate hard exit the final exception is best-effort — the same guarantee the Android SDK gives. Services that care should keep flushAt low.

💚 How did you test it?

• 15 new tests: 7 in the core PostHogErrorTrackingAutoCaptureIntegrationTest (local-only gate installs/refuses, CaptureTarget capture+flush, capture-not-suppressed-by-dedup, marking for later log mirrors, JVM stderr fallback, non-owning instance cannot tear down the installed handler), 2 in PostHogCapturedThrowablesTest (identity keying, value-equal throwables both captured), and 6 in the new server PostHogUncaughtExceptionTest (off by default, install+chain+restore on close, fatal/unhandled/mechanism assertions over a real /batch request, repeated setup keeps ownership, second client does not stack a handler, works with no remote config).
• ./gradlew :posthog:test :posthog-server:test pass (PostHogErrorTrackingAutoCaptureIntegrationTest 33 tests, PostHogCapturedThrowablesTest 2, PostHogUncaughtExceptionTest 6), plus :posthog:apiCheck / :posthog-server:apiCheck and spotlessCheck. API dumps regenerated; additive only.

📝 Checklist

• I reviewed the submitted code.
• I added tests to verify the changes.
• I updated the docs if needed.
• No breaking change or entry added to the changelog.

If releasing new changes

• Ran pnpm changeset to generate a changeset file
• Added the "release" label to the PR to indicate we're publishing new versions for the affected packages

🔗 Stacked PR

Position 3 of 4. Base: cat/java-et-server-config (PR #670).

1. PR feat(error-tracking): complete exception chain metadata and in-app classification #669 — core exception chain metadata + in-app classification
2. PR feat(server): expose error tracking config and captureException options #670 — server error-tracking config and captureException options
3. this PR — opt-in server uncaught-exception capture
4. cat/java-et-logback — new posthog-server-logback appender module