Centralize seat capacity accounting and the payment provider registry.

Replace manualProviders.ts with a paymentProviders.ts registry
(automatic vs manual settlement) and move all seat counting into
capacity.ts as the single source of truth: only paid/checked-in tickets
and pending_approval payments hold a seat, so abandoned checkouts never
block sales. Admins can now knowingly approve a payment over capacity
(allowOverCapacity), with the booking and admin UIs updated to match.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

frontend/src/lib/api/payments.ts

@@ -1,6 +1,14 @@
 import { fetchApi } from './client';
 import type { Payment, PaymentWithDetails } from './types';
 
+// Mirrors backend/src/lib/paymentProviders.ts: manual gateways need an admin to
+// verify the money arrived; automatic ones (lightning) confirm themselves.
+export const MANUAL_PAYMENT_PROVIDERS = ['tpago', 'bank_transfer', 'card', 'cash'];
+
+export function isManualProvider(provider: string): boolean {
+  return MANUAL_PAYMENT_PROVIDERS.includes(provider);
+}
+
 export const paymentsApi = {
   getAll: (params?: { status?: string; provider?: string; pendingApproval?: boolean; eventId?: string; eventIds?: string[] }) => {
     const query = new URLSearchParams();
@@ -21,10 +29,10 @@ export const paymentsApi = {
       body: JSON.stringify(data),
     }),
 
-  approve: (id: string, adminNote?: string, sendEmail: boolean = true) =>
+  approve: (id: string, adminNote?: string, sendEmail: boolean = true, allowOverCapacity: boolean = false) =>
     fetchApi<{ payment: Payment; message: string }>(`/api/payments/${id}/approve`, {
       method: 'POST',
-      body: JSON.stringify({ adminNote, sendEmail }),
+      body: JSON.stringify({ adminNote, sendEmail, allowOverCapacity }),
     }),
 
   reject: (id: string, adminNote?: string, sendEmail: boolean = true) =>