Using Rules, you can easily prevent the appearance of a flat rate shipping service on the checkout form based on the order total. We supply a condition in the Commerce Price group called Price comparison that makes it simply to check the total of an order. Simply add that condition to the rules component governing the availability of your delivery flat rate service, choose the order total using the commerce-line-item:order:commerce-order-total token, and then configure the remaining parameters to ensure the price is the right amount. (In your case I'd check to make sure the price is >= $10.00, since these components govern availability, not restrict it.)

What this will do is actually prevent the illegitimate option from appearing in the first place. Generally speaking, it's better customer experience to not show an option that cannot actually be selected. In the middle of the checkout process, you don't want a customer leaving the form for any reason at all - even if you think it's an upsell opportunity.

Much better to inform them of the delivery policy ahead of time and only make it available on the checkout form if they have passed the order total threshold. If I were to do anything, I'd add a message to the top of the shopping cart page or maybe the first checkout page stating that orders of $10 or more can be delivered with some sort of tasteful hint about other related products.

Free shipping is always a great upsell opportunity, but I wouldn't want to make my customer feel like they're missing out right when they're about to give me money. They're more likely to run off and buy from somewhere else.