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@Gordon
I'd stick the ferrite on the end nearest the Pico. I can't promise anything but it's worth a try...
I have a really similar setup for my bathroom lights (4 relays of that type, and then a 1M long wire that goes to the light pull) and I was amazed at the amount of noise it received when using the internal 40kOhm pullup resistors - I had to move to much lower values to get it to work reliably and not turn on and off all the time :)
I'd definitely power the shift register from 5v - that should help a lot too. The way the relays usually are (pull down to turn on) it means that if you power them from 5v but feed them with a 3.3v signal, the drivers are 'half on' all the time, because 3.3v is still less than the 5v they're expecting to be turned off.