BeatStep will keep those settings to itself after configured through MCC, just like all the other Arturia controllers
Simply use the "Store" button, to send the config to a specific "Memory" (i.e. config slot)
All you need to do is find out what are the correct notes the SP404sx is expecting per its own pads.
Maybe you can even access the other banks. Check the SP404sx MIDI Implementation charts and see if there's a Specific CC or Bank+Patch/Program Change that triggers that or if other banks are accessible on other MIDI channels, then you'll be able to have various BeatStep configs for each channel or a 2nd config with BeatStep's pads sending a specific patch change and then you recall back to your usual config for the pads with notes out.
BeatStep's configurations are quite versatile enough to explore these options