feat: settings modal with database export and import

Adds a gear button to the nav that opens a settings modal with:
- Export: GET /api/export returns all instances as a JSON backup file
  with a Content-Disposition attachment header
- Import: POST /api/import validates and bulk-replaces all instances;
  client uses FileReader to POST the parsed JSON, with a confirm dialog
  before destructive replace

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

index.html

@@ -22,6 +22,7 @@
       <span class="nav-divider">·</span>
       <span id="nav-version"></span>
     </div>
+    <button class="nav-btn" onclick="openSettingsModal()" title="Settings">&#9881;</button>
   </nav>
 
   <main>
@@ -171,6 +172,31 @@
   </div>
 </div>
 
+<!-- SETTINGS MODAL -->
+<div id="settings-modal" class="modal-overlay">
+  <div class="modal">
+    <div class="modal-header">
+      <span class="modal-title">Settings</span>
+      <button class="modal-close" onclick="closeSettingsModal()">&#x2715;</button>
+    </div>
+    <div class="modal-body">
+      <div class="settings-section">
+        <div class="settings-section-title">Export</div>
+        <p class="settings-desc">Download all instance data as a JSON backup file.</p>
+        <button class="btn btn-secondary" onclick="exportDB()">Export Database</button>
+      </div>
+      <div class="settings-section">
+        <div class="settings-section-title">Import</div>
+        <p class="settings-desc">Restore from a backup file. This replaces all current instances.</p>
+        <div class="import-row">
+          <input type="file" id="import-file" accept=".json" class="form-input import-file-input">
+          <button class="btn btn-danger" onclick="importDB()">Import</button>
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
 <!-- TOAST -->
 <div class="toast" id="toast">
   <div class="toast-dot"></div>