I've walked the CF Sept. 15-Oct 15 (2016) with no sleeping bag and just a liner (and a down jacket to wear in the evening and could sleep in if I hit a cold night). Did not regret that decision. That said, I've also walked the CP with a silk liner (4oz, Cocoon Expedition Silk) for when it was warmer (and it was) plus a very light backpacking quilt for when it was cooler (and it was) the last two weeks of October 2017. Did not regret that decision either. Your time frame, I'd probably not carry the quilt. But HYOH.
If you decide to take something, my sincere recommendation is a backpacking quilt (the lightest possible--this is not camping after all--walls and sleeping pilgrims stacking in bunk beds keep even unheated alburgues much warmer than the outside air). I use Enlightened Equipment 50 degree (12oz for me, 8oz for my wife--they are made to size and she is much smaller than me), but also look at Hammock Gear (slightly cheaper, slightly heavier). Almost all backpackers these days carry quilts not sleeping bags Spring-to-Fall. The weight savings (no bottom to the bag (you lay on the bottom down and compress it to uselessness anyway, no hood) is well worth it. Order now because they have long lead times (these are small cottage manufacturers).
Buen Camino,
Jo Jo