perf(coderd/x/chatd): add process-wide config cache for hot DB queries (#23272)

## Summary

Adds a process-wide cache for three hot database queries in `chatd` that
were hitting Postgres on **every chat turn** despite returning
rarely-changing configuration data:

| Query | Before (50k turns) | After | Reduction |
|---|---|---|---|
| `GetEnabledChatProviders` | ~98.6k calls | ~500-1000 | ~99% |
| `GetChatModelConfigByID` | ~49.2k calls | ~500-1000 | ~98% |
| `GetUserChatCustomPrompt` | ~46.7k calls | ~1000-2000 | ~97% |

These were identified via `coder exp scaletest chat` (5000 concurrent
chats × 10 turns) as the dominant source of Postgres load during chat
processing.

## Design

Follows the established **webpush subscription cache pattern**
(`coderd/webpush/webpush.go`):
- `sync.RWMutex` + `tailscale.com/util/singleflight` (generic) +
generation-based stale prevention + TTL
- 10s TTL for provider/model config, 5s TTL for user prompts
- Negative caching for `sql.ErrNoRows` on user prompts (the common case
— most users don't set custom prompts)
- Deep-clones `ChatModelConfig.Options` (`json.RawMessage` = `[]byte`)
on both store and read paths

### Invalidation

Single pubsub channel (`chat:config_change`) with kind discriminator for
cross-replica cache invalidation. Seven publish points in
`coderd/chats.go` cover all admin mutation endpoints
(create/update/delete for providers and model configs, put for user
prompts).

_This PR was generated with mux and was reviewed by a human_

coderd/pubsub/chatconfigevent.go

@@ -0,0 +1,52 @@
+package pubsub
+
+import (
+	"context"
+	"encoding/json"
+
+	"github.com/google/uuid"
+	"golang.org/x/xerrors"
+)
+
+// ChatConfigEventChannel is the pubsub channel for chat config
+// changes (providers, model configs, user prompts). All replicas
+// subscribe to this channel to invalidate their local caches.
+const ChatConfigEventChannel = "chat:config_change"
+
+// HandleChatConfigEvent wraps a typed callback for ChatConfigEvent
+// messages, following the same pattern as HandleChatEvent.
+func HandleChatConfigEvent(cb func(ctx context.Context, payload ChatConfigEvent, err error)) func(ctx context.Context, message []byte, err error) {
+	return func(ctx context.Context, message []byte, err error) {
+		if err != nil {
+			cb(ctx, ChatConfigEvent{}, xerrors.Errorf("chat config event pubsub: %w", err))
+			return
+		}
+		var payload ChatConfigEvent
+		if err := json.Unmarshal(message, &payload); err != nil {
+			cb(ctx, ChatConfigEvent{}, xerrors.Errorf("unmarshal chat config event: %w", err))
+			return
+		}
+
+		cb(ctx, payload, err)
+	}
+}
+
+// ChatConfigEvent is published when chat configuration changes
+// (provider CRUD, model config CRUD, or user prompt updates).
+// Subscribers use this to invalidate their local caches.
+type ChatConfigEvent struct {
+	Kind ChatConfigEventKind `json:"kind"`
+	// EntityID carries context for the invalidation:
+	//   - For providers: uuid.Nil (all providers are invalidated).
+	//   - For model configs: the specific config ID.
+	//   - For user prompts: the user ID.
+	EntityID uuid.UUID `json:"entity_id"`
+}
+
+type ChatConfigEventKind string
+
+const (
+	ChatConfigEventProviders   ChatConfigEventKind = "providers"
+	ChatConfigEventModelConfig ChatConfigEventKind = "model_config"
+	ChatConfigEventUserPrompt  ChatConfigEventKind = "user_prompt"
+)