I also vote for bringing your pillow from home. Even a small improvement in quality of sleep would overcome the weight penalty of hauling it during the day. Almost nothing is more important to day-after-day cycling performance than a good night's sleep. At least that is my experience.
I bought 3 smallish and light weight but not very compressible pillows from IKEA for when my family tours. This was after trying a number of other options (clothes bags, inflatable, etc.). They work well for us, but I would haul full sized pillows if needed.
Sleep well.
Jim