What you are describing is impossible to do, the cookie will be set new on another domain and you will not be able to hendle it correctly. Your sGTM does not set cookies on a third-party domain at all. Cookies set on your domain are not actually available on the third-party domain (you will not see them at all in Safari).
The only correct option here is to use checkout on your domain. If this is not possible - then return the user after successful payment to the thank you page on your domain where the events will work, this option is available usually on any third-party checkout.