Is money earned from working in a private institute, where doubt is cast upon its accreditation and the validity of its certificates, considered unlawful money (haram)?

There is no objection to joining an institute that teaches something permissible, whether accredited or not accredited. It is permissible to promote it, advertise it, and receive payment for it. However, one must not deceive people regarding the institute's accreditation status. The permissibility of this is further confirmed by what the questioner mentioned about his mother's anger towards him when he left his job, as obedience to parents in matters of doubt is obligatory according to most scholars.