feat(doctor): add custom types check and centralize BeadsCustomTypes

- Add BeadsCustomTypes constant ("agent,role,rig,convoy,slot") to avoid
  hardcoded strings scattered across the codebase
- Add CustomTypesCheck to gt doctor that verifies Gas Town custom types
  are registered with beads, with --fix support
- Register custom types during gt init (best-effort, skips if no beads)
- Update install.go, rig_check.go, and rig/manager.go to use the constant

This ensures consistent type registration across all code paths and
catches misconfigured beads databases via gt doctor.

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

internal/cmd/init.go

@@ -3,10 +3,12 @@ package cmd
 import (
 	"fmt"
 	"os"
+	"os/exec"
 	"path/filepath"
 	"strings"
 
 	"github.com/spf13/cobra"
+	"github.com/steveyegge/gastown/internal/constants"
 	"github.com/steveyegge/gastown/internal/git"
 	"github.com/steveyegge/gastown/internal/rig"
 	"github.com/steveyegge/gastown/internal/style"
@@ -80,6 +82,16 @@ func runInit(cmd *cobra.Command, args []string) error {
 		fmt.Printf("   ✓ Updated .git/info/exclude\n")
 	}
 
+	// Register custom beads types for Gas Town (agent, role, rig, convoy, slot).
+	// This is best-effort: if beads isn't installed or DB doesn't exist, we skip.
+	// The doctor check will catch missing types later.
+	if err := registerCustomTypes(cwd); err != nil {
+		fmt.Printf("   %s Could not register custom types: %v\n",
+			style.Dim.Render("⚠"), err)
+	} else {
+		fmt.Printf("   ✓ Registered custom beads types\n")
+	}
+
 	fmt.Printf("\n%s Rig initialized with %d directories.\n",
 		style.Bold.Render("✓"), created)
 	fmt.Println()
@@ -127,3 +139,34 @@ func updateGitExclude(repoPath string) error {
 	// Write back
 	return os.WriteFile(excludePath, append(content, []byte(additions)...), 0644)
 }
+
+// registerCustomTypes registers Gas Town custom issue types with beads.
+// This is best-effort: returns nil if beads isn't available or DB doesn't exist.
+// Handles gracefully: beads not installed, no .beads directory, or config errors.
+func registerCustomTypes(workDir string) error {
+	// Check if bd command is available
+	if _, err := exec.LookPath("bd"); err != nil {
+		return nil // beads not installed, skip silently
+	}
+
+	// Check if .beads directory exists
+	beadsDir := filepath.Join(workDir, ".beads")
+	if _, err := os.Stat(beadsDir); os.IsNotExist(err) {
+		return nil // no beads DB yet, skip silently
+	}
+
+	// Try to set custom types
+	cmd := exec.Command("bd", "config", "set", "types.custom", constants.BeadsCustomTypes)
+	cmd.Dir = workDir
+	output, err := cmd.CombinedOutput()
+	if err != nil {
+		// Check for common expected errors
+		outStr := string(output)
+		if strings.Contains(outStr, "not initialized") ||
+			strings.Contains(outStr, "no such file") {
+			return nil // DB not initialized, skip silently
+		}
+		return fmt.Errorf("%s", strings.TrimSpace(outStr))
+	}
+	return nil
+}