Looks like you can make this work using the EnumMember attribute (found in System.Runtime.Serialization).
public enum PermissionType
{
[EnumMember(Value = "can_fly")]
PermissionToFly,
[EnumMember(Value = "can_swim")]
PermissionToSwim
}
If you use those attributes you should also not need to set the ItemConverterType in the JsonProperty attribute on the list.