In each case, I think the meanings would be very similar in your examples. However, I might say "handle it with little effort" just to avoid having "with" twice so close together. Perhaps "cope with" is more passive and only suggests a lack of problems, while "handle" is more active and "deal with" is somewhere between the two.
In some cases they are not interchangeable though. For example:
“I will deal with/ handle this customer” You would not say “cope with” here.