As far as we know, it is not possible to become a verified merchant on Pinterest if you are an Etsy seller. [read the guidelines here] This is because one – Etsy is a marketplace and not your own domain, and two real-time sync for inventory is tough as it needs to happen automatically via a platform like Shopify to be eligible for verification as a merchant.
You can read more in the Pinterest community thread here.In order to promote your Etsy store, you will have to manually upload your pins as product pins and tag them with the relevant information as well as keywords. You will find a lesson in Phase 6 on how to use product pins.
If your goal is to have a verified merchant account, we would encourage you to in the long-run open a Shopify store once you have enough demand and sales for your products or keep an eye out for Pinterest changes to their program.