Hi Charlie,
I did an AS 1700 myself not long ago, and I also bought a new spring, which is much easier than reusing the old one. You'll find that that holder it's supplied in is slightly smaller in diameter than the inside of the barrel. Just lay it on top (check the direction!) and push the spring carefully down into the barrel. As the spring escapes from the holder it expands into the barrel.The grease you put on the inside wall beforehand will be squeezed outwards rather than downwards.
I wasn't sure how to describe the right amount of grease, so I looked for a video on you tube and found this one - Classic Watchmaking: Barrel Wall Lubrication. He greases the barrel wall right at the end. The answer it seems is a HUGE amount! Certainly much, much more than I put in. Maybe some of our more experienced members can offer their opinions.