You basically have two options here:

1. Disable the Cart module entirely. In this scenario, you would create an order on the backend in the Checkout: Checkout status and then send your customers a link directly to the checkout form. URLs for checkout would be dependent on the Order ID, so for Order 50, the URL you'd send would be http://www.example.com/checkout/50.
2. Keep the Cart module enabled, but edit the shopping cart form and remove the quantity edit / line item delete fields from the View. It's built entirely using Views, so you can remove these elements with no side effect. The only hitch is that if you remove the form elements from the View, the shopping cart form will no longer render as a form - Views only does that when form elements are present in the Views field list. You'd need to have a footer link or some other element on the page that could send the customer on to the appropriate checkout URL using the pattern mentioned above. If you put the Order ID in the View as a hidden field, you should be able to use it as a token in text area handlers in regions of the View like the footer.