There is no native API to simulate button presses on a joystick.
For keyboard and mouse, all HID devices that register as either of those devices "feed" a single tracked state, which can be altered simply by telling Windows that said state should change.
Joysticks on the other hand require a custom driver that fakes an entire device being present, which is much more complex.
Does this mean I *have* to map the game commands to keypresses and use those instead of being able to use the joystick buttons?
Currently, yes. However, as even the smallest keyboard should have at least 78 keys, with a number of modifiers(Shift, Ctrl, Alt, or even RShift, RCtrl, and RAlt, if your game supports them), I would assume that's workable.