What is the secondary soil stack for; does it just have waste
connections from the kitchen (ie no toilet?) Could any such pipes be
run into a suitable surface gully outside the kitchen?

Sorry, I may have misled a little. The smaller pipe is not a soil stack as such, it is just a vent pipe connected to an access chamber, where the main sewer from the toilet and bathroom meet the connection from the kitchen drainage. The only other thing going into this are the rain water drains for that side of the house.

I must admit this is an education, I never knew the purpose of the
venting in sewage systems.