Unfortunately the API currently does not provide a way to change programs with a partial set of parameters. You have to provide all parameters, otherwise it will end up change the program incorrectly.
What the app does is to construct an HTTP GET command with all the required parameters, and send that to the controller. Because it has to load the program data in order to display the program anyways, it knows all the required parameters.
So if you want to use the existing API, you need to issue an /jp to obtain program data first, then /cp to change program, even if all you want to do is to set/reset program enable bit.
Alternatively, if you don’t mind re-compiling the firmware, you can add a new HTTP GET command, say, call it /ep (enable program) which allows you to quickly enable/disable a program. It’s pretty easy to follow the existing HTTP GET command to create a new command.