Do you have any research to back up your claim that the buyers “don’t care” about the handling fee? They have no choice but to pay it!
Research as well as conventional wisdom dictates that customers do care, that people don’t like being nickel and dimed, hence the ‘Free Shipping’ promotions many online retailers use. Amazon even offer a subscription service with the major perk being no delivery charges.
As for whether price partitioning directly affects sales, according my understanding of current research it depends on many factors such as your type of customers, customers’ goals etc
Another topic to bear in mind is the perceived “fairness” of partitioned prices. Charging shipping fees that are unknown at the time of the order, but are charged later based on the true price paid (e.g: to the shipping courier) would seem “fair” and should appeal to customers. At the other extreme, charges such as an “Intellectual Property protection fee” which is perceived as unfair because it is not a direct tax or fee paid by a company can affect customer satisfaction and customer attitudes toward the company.
A good customer relationship and customer loyalty is priceless. With it, customers will never leave your brand come what may. Loose it and customers will jump ship as soon as a viable alternative pops up.