feat: Add /sync-bank endpoint to trigger bank sync and inference from web UI

Adds a new GET /sync-bank route that runs sync_to_sheets (2026) + infer_payments + flush_cache,
capturing all output and displaying it on a styled results page. Adds "Tools: [Sync Bank Data]"
nav link to all templates.

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

templates/adults.html

@@ -464,6 +464,10 @@
             <a href="/reconcile-juniors">[Junior Payment Reconciliation]</a>
             <a href="/payments">[Payments Ledger]</a>
         </div>
+        <div class="nav-archived">
+            <span style="color: #666; margin-right: 5px;">Tools:</span>
+            <a href="/sync-bank">[Sync Bank Data]</a>
+        </div>
     </div>
 
     <h1>Adults Dashboard</h1>
@@ -503,7 +507,7 @@
                         {{ cell.text }}
                         {% if cell.status == 'unpaid' or cell.status == 'partial' %}
                         <button class="pay-btn"
-                            onclick="showPayQR('{{ row.name|e }}', {{ cell.amount }}, '{{ cell.month|e }}')">Pay</button>
+                            onclick="showPayQR('{{ row.name|e }}', {{ cell.amount }}, '{{ cell.month|e }}', '{{ cell.raw_month }}')">Pay</button>
                         {% endif %}
                     </td>
                     {% endfor %}
@@ -511,7 +515,7 @@
                         {{ "%+d"|format(row.balance) if row.balance != 0 else "0" }}
                         {% if row.balance < 0 %}
                         <button class="pay-btn"
-                            onclick="showPayQR('{{ row.name|e }}', {{ -row.balance }}, '{{ row.unpaid_periods|e }}')">Pay All</button>
+                            onclick="showPayQR('{{ row.name|e }}', {{ -row.balance }}, '{{ row.unpaid_periods|e }}', '{{ row.raw_unpaid_periods|e }}')">Pay All</button>
                         {% endif %}
                     </td>
                 </tr>
@@ -857,9 +861,13 @@
                 showMemberDetails(nextName);
             }
         }
-        function showPayQR(name, amount, month) {
+        function showPayQR(name, amount, month, rawMonth) {
             const account = "{{ bank_account }}";
-            const message = `${name} / ${month}`;
+            // Convert YYYY-MM to MM/YYYY for infer_payments.py compatibility
+            const numericMonth = rawMonth.includes('+')
+                ? rawMonth.split('+').map(p => p.replace(/(\d{4})-(\d{2})/, '$2/$1')).join('+')
+                : rawMonth.replace(/(\d{4})-(\d{2})/, '$2/$1');
+            const message = `${name} / ${numericMonth}`;
             const qrTitle = document.getElementById('qrTitle');
             const qrImg = document.getElementById('qrImg');
             const qrAccount = document.getElementById('qrAccount');