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/exp_chats.go

@@ -33,6 +33,7 @@ import (
 	"github.com/coder/coder/v2/coderd/database"
 	"github.com/coder/coder/v2/coderd/database/db2sdk"
 	"github.com/coder/coder/v2/coderd/database/dbauthz"
+	dbpubsub "github.com/coder/coder/v2/coderd/database/pubsub"
 	"github.com/coder/coder/v2/coderd/externalauth"
 	"github.com/coder/coder/v2/coderd/externalauth/gitprovider"
 	"github.com/coder/coder/v2/coderd/httpapi"
@@ -110,6 +111,28 @@ func maybeWriteLimitErr(ctx context.Context, rw http.ResponseWriter, err error)
 	return false
 }
 
+func publishChatConfigEvent(logger slog.Logger, ps dbpubsub.Pubsub, kind pubsub.ChatConfigEventKind, entityID uuid.UUID) {
+	payload, err := json.Marshal(pubsub.ChatConfigEvent{
+		Kind:     kind,
+		EntityID: entityID,
+	})
+	if err != nil {
+		logger.Error(context.Background(), "failed to marshal chat config event",
+			slog.F("kind", kind),
+			slog.F("entity_id", entityID),
+			slog.Error(err),
+		)
+		return
+	}
+	if err := ps.Publish(pubsub.ChatConfigEventChannel, payload); err != nil {
+		logger.Error(context.Background(), "failed to publish chat config event",
+			slog.F("kind", kind),
+			slog.F("entity_id", entityID),
+			slog.Error(err),
+		)
+	}
+}
+
 // EXPERIMENTAL: this endpoint is experimental and is subject to change.
 func (api *API) watchChats(rw http.ResponseWriter, r *http.Request) {
 	ctx := r.Context()
@@ -3052,6 +3075,8 @@ func (api *API) putUserChatCustomPrompt(rw http.ResponseWriter, r *http.Request)
 		return
 	}
 
+	publishChatConfigEvent(api.Logger, api.Pubsub, pubsub.ChatConfigEventUserPrompt, apiKey.UserID)
+
 	httpapi.Write(ctx, rw, http.StatusOK, codersdk.UserChatCustomPrompt{
 		CustomPrompt: updatedConfig.Value,
 	})
@@ -3886,6 +3911,8 @@ func (api *API) createChatProvider(rw http.ResponseWriter, r *http.Request) {
 		}
 	}
 
+	publishChatConfigEvent(api.Logger, api.Pubsub, pubsub.ChatConfigEventProviders, uuid.Nil)
+
 	httpapi.Write(
 		ctx,
 		rw,
@@ -3972,6 +3999,8 @@ func (api *API) updateChatProvider(rw http.ResponseWriter, r *http.Request) {
 		return
 	}
 
+	publishChatConfigEvent(api.Logger, api.Pubsub, pubsub.ChatConfigEventProviders, uuid.Nil)
+
 	httpapi.Write(
 		ctx,
 		rw,
@@ -4026,6 +4055,8 @@ func (api *API) deleteChatProvider(rw http.ResponseWriter, r *http.Request) {
 		return
 	}
 
+	publishChatConfigEvent(api.Logger, api.Pubsub, pubsub.ChatConfigEventProviders, uuid.Nil)
+
 	rw.WriteHeader(http.StatusNoContent)
 }
 
@@ -4205,6 +4236,8 @@ func (api *API) createChatModelConfig(rw http.ResponseWriter, r *http.Request) {
 		}
 	}
 
+	publishChatConfigEvent(api.Logger, api.Pubsub, pubsub.ChatConfigEventModelConfig, inserted.ID)
+
 	httpapi.Write(ctx, rw, http.StatusCreated, convertChatModelConfig(inserted))
 }
 
@@ -4376,6 +4409,8 @@ func (api *API) updateChatModelConfig(rw http.ResponseWriter, r *http.Request) {
 		}
 	}
 
+	publishChatConfigEvent(api.Logger, api.Pubsub, pubsub.ChatConfigEventModelConfig, updated.ID)
+
 	httpapi.Write(ctx, rw, http.StatusOK, convertChatModelConfig(updated))
 }
 
@@ -4416,6 +4451,8 @@ func (api *API) deleteChatModelConfig(rw http.ResponseWriter, r *http.Request) {
 		return
 	}
 
+	publishChatConfigEvent(api.Logger, api.Pubsub, pubsub.ChatConfigEventModelConfig, modelConfigID)
+
 	rw.WriteHeader(http.StatusNoContent)
 }