Kate,
We've walked the CF (Sept 15-Oct 15) with just a liner bag. But we also carry very light down jackets (for the evenings), which we slept in a couple of times when it got chilly and the pilgrims were not packed in close enough to generate enough heat (and there was no blankets. Pro tip: ask for blankets; sometimes the alburgues have them, but not right on the beds). Look at Montbell for good, ultralight jackets.
We need to talk about those microfiber liner bags. That stuff is heavy. Most I see are over a pound (20 oz or so). Ridiculously heavy. That is the weight of 20-degree backpacking quilt I use for fall in the American mountains.
For less than the weight of a microfleece lince, I can carry a silk liner bag (4oz, Cocoon expedition) AND a down backpacking quilt (50 degree (more than enough) 12 oz, Enlightened Equipment), that is a far more flexible system (covering a greater range of possible temperatures--it was hot in Sept. 2016). Personally, I think the backpacking quilt would be overkill for Sept (it was necessary in the end of October on the CP) and would just go with the down jacket, but the point is you could do that and still save a quarter pound off the microfiber fleece.
Buen Camino,
Jo Jo