feat(docker): add a folder picker to both path fields

Both paths in the adoption form had to be typed. Each gets a Browse button
using the same path-input-group markup Link Existing uses, so the two look and
behave alike.

What they can browse differs, and that is the point. The host workspace path
reuses the existing host picker. The container workdir cannot: an adopted
container has nothing mounted at a matching host path, so a host listing would
be a different filesystem — and getting this field wrong is the source of the
opaque OCI chdir error at launch, which makes it the field that most needs to
be clickable.

Adds a read-only POST /api/docker-cases/browse: one `ls` through docker exec, no
writes, no lifecycle, path shell-escaped like every other value. `ls -Ap` marks
directories with a trailing slash and keeps names with spaces intact.

PathPicker takes an optional fetchListing source rather than being forked: the
container variant only swaps where the rows come from, and reuses the rendering,
navigation, Up and Choose/Select unchanged.

src/docker-hosts.ts

@@ -1239,6 +1239,60 @@ export async function probeAdoptableContainer(
   }
 }
 
+/** One directory listing from INSIDE a container, shaped like the host picker's. */
+export interface DockerBrowseResult {
+  path: string;
+  parent: string | null;
+  entries: Array<{ name: string; path: string; type: 'directory' | 'file' }>;
+  error?: string;
+}
+
+/**
+ * List a directory INSIDE a container, for the adoption form's container-workdir
+ * picker. The host filesystem picker cannot serve this: the path lives in the
+ * container, and for an adopted container nothing is mounted at a matching host
+ * location, so the user would otherwise be typing a path blind.
+ *
+ * Read-only: one `ls` through `docker exec`, no writes, no lifecycle. The path
+ * is shell-escaped like every other value this module interpolates, and output
+ * is parsed as NUL-free lines with a leading type marker so a filename with
+ * spaces survives.
+ */
+export async function browseInContainer(
+  docker: Pick<SessionDocker, 'engine' | 'context' | 'daemonHost' | 'containerName'>,
+  path: string
+): Promise<DockerBrowseResult> {
+  const target = path && path.startsWith('/') ? path : '/';
+  const parent = target === '/' ? null : target.replace(/\/+$/, '').split('/').slice(0, -1).join('/') || '/';
+  if (IS_TEST_MODE) return { path: target, parent, entries: [] };
+  const argv = dockerEngineArgv(docker);
+  // `-p` marks directories with a trailing slash; `-A` shows dotfiles but not
+  // the . and .. entries the picker navigates with its own Up control.
+  const script = `cd ${shellescape(target)} 2>/dev/null && ls -Ap 2>/dev/null || echo __ERR__`;
+  try {
+    const { stdout } = await execFileAsync(
+      argv[0],
+      [...argv.slice(1), 'exec', docker.containerName, 'sh', '-lc', script],
+      { timeout: DOCKER_PROBE_TIMEOUT_MS, maxBuffer: 4 * 1024 * 1024 }
+    );
+    if (stdout.includes('__ERR__')) return { path: target, parent, entries: [], error: 'Not a readable directory' };
+    const base = target.endsWith('/') ? target : `${target}/`;
+    const entries = stdout
+      .split('\n')
+      .map((line) => line.trim())
+      .filter(Boolean)
+      .map((name) => {
+        const isDir = name.endsWith('/');
+        const clean = isDir ? name.slice(0, -1) : name;
+        return { name: clean, path: `${base}${clean}`, type: (isDir ? 'directory' : 'file') as 'directory' | 'file' };
+      })
+      .sort((a, b) => Number(b.type === 'directory') - Number(a.type === 'directory') || a.name.localeCompare(b.name));
+    return { path: target, parent, entries };
+  } catch (err) {
+    return { path: target, parent, entries: [], error: err instanceof Error ? err.message : String(err) };
+  }
+}
+
 /**
  * Resolve the host's IP on the default docker bridge (the address a container
  * reaches as `host.docker.internal`), so the server can bind a hooks-only listener