You can't customize checkout page in shopify.
Why does the amount need to be sent from cart to checkout page? I'm interested to knowing the needed behind that.
Since this is Shipping related would you not need the customers Shipping information first? That's added in Step 1 of the checkout.
Have you looked at the CarrierService API? That would let you build a custom app to surface your rates to the Customer.
There's also a bunch of apps that use the CarrierService API as well so might be something useful ready to be used now.