Branching off underground rainwater pipe afterwards

The magic word is "slip coupling" You cut out enough pipe material so that a branch with three couplings fits and an approximately 30cm long spigot-to-spigot sleeve piece (which you can get from the cut-out pipe minus the branch length). Then you assemble the branch onto the existing pipeline and slide the slip coupling onto the sleeve piece or the other end of the existing pipeline, and when you have the new section (branch + sleeve piece) in between, you gently tap the slip coupling centrally onto the new-old transition.