How long is the strip, what sort of LEDs? What's the pinout of the strip? Is it individual R/G/B channels with a common connection- is it a common power or ground connection?
The ULN2003A is inverting by nature, and the SN3218 is a sinking, constant current driver. I'd say they would make a poor combination but I understand from your example what you're trying to accomplish.
If your LEDs were common ground/cathode, it would probably be easier to ditch the SN3218 and use just the ULN2003A, in combination with software PWM on the Pi. Or possibly even something like Pi Blaster- https://github.com/sarfata/pi-blaster