After waiting for the asset to load, using UEditorAssetSubsystem::RenameAsset worked.
FAssetRegistryModule& AssetRegistryModule = FModuleManager::LoadModuleChecked<FAssetRegistryModule>("AssetRegistry");
if (AssetRegistryModule.Get().IsLoadingAssets())
{
AssetRegistryModule.Get().OnFilesLoaded().AddRaw(this, &RenameAsset);
return;
}
RenameAsset();
void RenameAsset()
{
auto editorAssetSubsystem = GEditor->GetEditorSubsystem<UEditorAssetSubsystem>();
editorAssetSubsystem->RenameAsset(oldPath, newPath);
}