Support Supabase branches (#1394)

<!-- This is an auto-generated description by cubic. -->

## Summary by cubic
Adds Supabase database branch selection per app, with a new schema field
and UI to choose a branch after connecting a project. Resets branch when
changing or disconnecting the project to keep state consistent.

- **New Features**
  - Added apps.supabase_branch_id column.
- Branch dropdown in SupabaseConnector shown after a project is
connected; selection persists and triggers app refresh.
- New state and hooks: supabaseBranchesAtom, loadBranches(projectId),
setAppBranch(branchId).
- IPC endpoints: supabase:list-branches and supabase:set-app-branch;
setting/unsetting project also clears the branch.

- **Migration**
- Apply drizzle migration 0013_supabase_branch.sql to add the
supabase_branch_id column (defaults to null).

<!-- End of auto-generated description by cubic. -->

<!-- CURSOR_SUMMARY -->
---

> [!NOTE]
> Adds Supabase database branch selection per app, including parent
project tracking, new IPC endpoints, UI dropdown, and an accompanying DB
migration with e2e tests.
> 
> - **Database**:
> - Add `apps.supabase_parent_project_id` via migration
`drizzle/0015_complete_old_lace.sql`; snapshot and journal updated.
> - **IPC/Main**:
> - New `supabase:list-branches` handler and management client
`listSupabaseBranches` (real API + test stubs).
> - Update `supabase:set-app-project` to accept `{ projectId,
parentProjectId?, appId }`; unset clears both IDs.
> - `get-app` resolves `supabaseProjectName` using
`supabase_parent_project_id` when present.
> - **Types & Client**:
> - Add `SupabaseBranch`, `SetSupabaseAppProjectParams`, and
`App.supabaseParentProjectId`; expose `listSupabaseBranches` and updated
`setSupabaseAppProject` in `ipc_client` and preload whitelist.
> - **UI/Hooks**:
> - Supabase UI: branch dropdown in `SupabaseConnector` with
`loadBranches`, selection persists via updated `setAppProject`.
> - State: add `supabaseBranchesAtom`; `useSupabase` gets `branches`,
`loadBranches`, new param shape for `setAppProject`.
>   - TokenBar/ChatInput: add `data-testid` for token bar and toggle.
> - **Supabase Context (tests)**:
> - Test build returns large context for `test-branch-project-id` to
validate branch selection.
> - **E2E Tests**:
> - Add `supabase_branch.spec.ts` and snapshot verifying branch
selection affects token usage.
> 
> <sup>Written by [Cursor
Bugbot](https://cursor.com/dashboard?tab=bugbot) for commit
33054278db8396b4371ed6e8224105cb5684b7ac. This will update automatically
on new commits. Configure
[here](https://cursor.com/dashboard?tab=bugbot).</sup>
<!-- /CURSOR_SUMMARY -->

src/supabase_admin/supabase_management_client.ts

@@ -168,6 +168,61 @@ export async function deleteSupabaseFunction({
   );
 }
 
+export async function listSupabaseBranches({
+  supabaseProjectId,
+}: {
+  supabaseProjectId: string;
+}): Promise<
+  Array<{
+    id: string;
+    name: string;
+    is_default: boolean;
+    project_ref: string;
+    parent_project_ref: string;
+  }>
+> {
+  if (IS_TEST_BUILD) {
+    return [
+      {
+        id: "default-branch-id",
+        name: "Default Branch",
+        is_default: true,
+        project_ref: "fake-project-id",
+        parent_project_ref: "fake-project-id",
+      },
+
+      {
+        id: "test-branch-id",
+        name: "Test Branch",
+        is_default: false,
+        project_ref: "test-branch-project-id",
+        parent_project_ref: "fake-project-id",
+      },
+    ];
+  }
+
+  logger.info(`Listing Supabase branches for project: ${supabaseProjectId}`);
+  const supabase = await getSupabaseClient();
+
+  const response = await fetch(
+    `https://api.supabase.com/v1/projects/${supabaseProjectId}/branches`,
+    {
+      method: "GET",
+      headers: {
+        Authorization: `Bearer ${(supabase as any).options.accessToken}`,
+      },
+    },
+  );
+
+  if (response.status !== 200) {
+    throw await createResponseError(response, "list branches");
+  }
+
+  logger.info(`Listed Supabase branches for project: ${supabaseProjectId}`);
+  const jsonResponse = await response.json();
+  return jsonResponse;
+}
+
 export async function deploySupabaseFunctions({
   supabaseProjectId,
   functionName,