How would I implement an interrupt that changes the duty cycle of an PWM output with the rising edge of the on-board button?
Can be rather simple or a bit more complicated. Just connect an isr-component to the pin and write an interrupt handler. Have a look into the "System Reference Guide" for CY_ISR and CY_ISR_PROTO and use isr_StartEx(YourHandler).
To be more complicated: Switches use to bounce, so a de-glitcher would be better.
And have a look into the schematics of your board, most switches connect to Gnd, so you'll have to look for a falling edge, not a rising one.
For debounce routines take a look at attached for SW solutions. HW is
precious on PSOC 4.
To change the PWM DC you use its PWM_WriteCompare function API,
see component datasheet for PWM.