Customizing Emergency SOS contacts for Family Apple Watch

Family Apple Watch Emergency SOS Contacts

I have kids with Watch SE2s, and was recently reconfiguring one of their watches (my new iPhone 17p broke the link). It looks like the Emergency Contacts for their watch is the same set of people that are on *my* emergency list (e.g., spouse, dad), rather than a separate list that would be applicable to their world (e.g. me, spouse). Is there a way to adjust this for the Family-managed watches?