Is it permissible to apply for a free American scholarship without mentioning obtaining a bachelor's degree that is not recognized in America, with the aim of improving one's living and professional situation?

If the granting body stipulates that the applicant must not have previously obtained a bachelor's degree, it is not permissible to conceal this from them; because it is forbidden deception and fraud. And the non-recognition of your degree in that country does not make concealment permissible. Rather, you must explain to them the reality of your studies and the non-recognition of your degree. If they grant you the scholarship after that, the matter is up to them.