Diagnose idle player battery usage

[kevinatlee]

Summary

• add an opt-in ?perf=1 diagnostic registry that survives join-to-room SPA navigation
• report actual Socket.IO/Engine.IO transport, connection, reconnect, upgrade, room-event, and packet counters
• count authoritative snapshot acceptance, exact duplicates, and stale/conflicting rejections without changing the decision path
• count App, RoomLobby, and LetterGrid renders with no state updates or recurring sampler
• provide manual Refresh/Copy Diagnostics controls without a polling timer
• document the idle-lobby CSS audit, transport hypothesis, and physical test matrix

Validation

• focused diagnostics/client/socket tests: 85 passed
• full suite: 819 passed
• privacy, formatting, lint, typecheck, build, and game-data verification
• dependency audit: 0 vulnerabilities
• production build and smoke test
• diff check

Physical gate

No battery optimization is included. Keep this PR in draft while the exact test image is measured for ten idle lobby minutes with ?perf=1, then capture the manually refreshed diagnostic snapshot.