Shopify maintains strict control over its checkout process for security so as far as I know it is not possible to integrate unapproved payment service provider.
You can consider to implement this approach that might work properly:
Create a Draft Order
Create a PayGreen payment link for the draft’s total and include the draft order ID as metadata
Redirect the customer to PayGreen’s hosted page
On PayGreen webhook success, verify signature, then complete the draft and mark it paid with an external transaction (“PayGreen”)
On failure/timeout, cancel the draft and restock