Anvil Studio may be more complex that you need, but yes, it should be able to do what you need.
AS will try to use a build in virtual synth to play back, but it should be able to send midi data to your keyboard instead.
There are a number of 'views' available, you should check them all to find out which one allows you best access to the PC (Program Change) items that are causing the problem.
The Staff view may not show them. The Piano Roll view may not do so by default, but you may need to select an option to show PC items. There is also a 'Midi Events' view that will show all the midi events.
One extra problem that was not mentioned before. There are two ways of selecting sounds/instruments/patches, depending on the keyboard/device. The PC (Program Change) command is the usual method, but this allows one byte only, which is fine for switching between - for example - the 128 GM patches available. More modern devices may well have a lot more sounds, divided between various 'Banks'. So you need more data to select these sounds, and this can vary between devices. There may well be a 'Bank Select' command needing two bytes (MSB and LSB - aka Most Significant Byte and Least Sig Byte) and this may then be followed by a PC command, or I think that for some devices the two bytes can select Bank and then sound within the bank. Once a specific Part/Channel has been set to a specific Bank, then subsequent PC commands may well change to other patches within the same bank, which might well NOT be what you want.
Check up in your manual for the keyboard regarding Bank Select and Program Change.
Geoff