This is a typical example of a stream pouring out of a Rock Garden. The rock is Oxforshire Limestone. The pool liner is always disguised like this for a 100% natural look.
A stream like this could also be extended to pour from the rock pool across the bottom of the garden and terminate in the "Folly" corner.
This rock though I feel could blend in better with the house, and also with the path and terrace, being much lighter than the brown Oxfordshire. But I would provide samples at a later stage for you/the clients to decide.
This is typical of my style whatever the rock, and shows a stream pouring out of a Rock Garden.
This again is the lighter coloured rock
Here's a smaller version of yours, with a stream and pool beside a path, that I created in Mulbery Walk (Kings Rd), and next is how the space looked before...