[cr138] Use BraveExtensionsBrowserClientImpl::Init to initialize resource manager

The resource manager is now initialized in
`ChromeExtensionsBrowserClient::Init`, so we override that function in order to
make use of it.

Chromium change:
https://source.chromium.org/chromium/chromium/src/+/efce54025758dc9d4f370c0269b332cb0d67474b

commit efce54025758dc9d4f370c0269b332cb0d67474b
Author: James Cook <jamescook@chromium.org>
Date:   Mon May 12 13:00:22 2025 -0700

    extensions: Use ChromeExtensionsBrowserClient on desktop Android, part 2

    This makes desktop Android extensions behavior more like Windows,
    Mac, and Linux.

    Delete DesktopAndroidExtensionsBrowserClient. Extract the parts of
    ChromeExtensionsBrowserClient that don't work on android to the file
    chrome_extensions_browser_client_non_android.cc

    Introduce an Init() method to UserScriptListener can be initialized
    after g_browser_process creation (otherwise it crashes).

    Fix a few small issues with ActivityLog initialization, prefs and
    some extensions utility functions.

    The CL is large, but most of it is cut-and-pasting functions from
    chrome_extensions_browser_client.cc to _non_android.cc.

    Bug: 415033270
    Change-Id: Icfcca2a69b523ced6b5471ba659a1440a303a5c0
    Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6509682
    Reviewed-by: Yaron Friedman <yfriedman@chromium.org>
    Commit-Queue: James Cook <jamescook@chromium.org>
    Reviewed-by: Devlin Cronin <rdevlin.cronin@chromium.org>
    Cr-Commit-Position: refs/heads/main@{#1459046}

browser/extensions/brave_extensions_browser_client_impl.cc

@@ -15,6 +15,11 @@ namespace extensions {
 
 BraveExtensionsBrowserClientImpl::BraveExtensionsBrowserClientImpl() {
   AddAPIProvider(std::make_unique<BraveExtensionsBrowserAPIProvider>());
+}
+
+void BraveExtensionsBrowserClientImpl::Init() {
+  ChromeExtensionsBrowserClient::Init();
+
   // ChromeComponentExtensionResourceManager's Data needs to be LazyInit'ed on
   // the UI thread (due to pdf_extension_util::AddStrings calling
   // g_browser_process->GetApplicationLocale() that has a DCHECK to that

browser/extensions/brave_extensions_browser_client_impl.h

@@ -18,6 +18,8 @@ class BraveExtensionsBrowserClientImpl : public ChromeExtensionsBrowserClient {
   BraveExtensionsBrowserClientImpl& operator=(
       const BraveExtensionsBrowserClientImpl&) = delete;
   ~BraveExtensionsBrowserClientImpl() override = default;
+
+  void Init() override;
 };
 
 }  // namespace extensions