You may not require to take registration only if you provide exempted service / supply exempted goods.
Zero rate transaction is different from exempted transaction. Export not comes under Exempt category. Hence, you will have to register under GST if your annual turnover exceeds 20 Lakhs.
With respect to Claiming Zero rate benefit on your Export transactions :
You have two options
Option 1 : You will have to Pay IGST on your own on your export transaction value and apply for Refund with Govt Or
Option 2 : You will have to provide Bond / Letter of Undertaking with Bank Guarantee to Juridicitional Commissioner / Assistant Commissioner and get it approved before export any tranasaction without Payment of IGST.