chore: move widget source into sidecar repo (widget-src/)

Widget builds from widget-src/ → public/widget.js
Vite outDir updated to ../public
.gitignore excludes node_modules

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

widget-src/tsconfig.json

@@ -0,0 +1,14 @@
+{
+  "compilerOptions": {
+    "target": "ES2020",
+    "module": "ESNext",
+    "moduleResolution": "bundler",
+    "jsx": "react-jsx",
+    "jsxImportSource": "preact",
+    "strict": true,
+    "esModuleInterop": true,
+    "outDir": "dist",
+    "skipLibCheck": true
+  },
+  "include": ["src"]
+}