change GET query string

If I understood you good, you have need to delete 'brand' segment because it's empty and it brings you to troubles BECAUSE it is empty.

But there must be a place where you handle with _GET array and there is nothing easier than unset all keys, where value is empty, before next process of these input data.

If you make this, you can simply ignore what is left in browser url address field and you don't need to modify it.