According to section 24 of CGST Act, 2017, persons making inter State taxable supply are compulsorily required to get registered. According to Sec 7 of IGST Act, 2017 export is an inter state supply. Therefore eventhough your turnover is less than 20 lakh you are required to get registered.
Zero rated supply is also a taxable supply as per law. It is not an exempted supply. Even if a particular goods or services is an exempted one, for the purpose of zero rated supply it will be taken as taxable supply. This is one of measures to promote export. So Registration is compulsory.