I have no experience with spider lilies, but looking at your images they remind me a little about the bulbs of Ornithogalum saundersiae in that the stems appear to leave a hole into the bulb when removed? I was a bit worried about that last winter when bringing them in for storage as some of the stems went mushy rather than drying out. However all of them were fine in the end and sprouted the following spring. I think there is a difference here when we have foliage that are affected by cold and natural decay as opposed to having a pathogen entering the foliage and eventually into the bulb.
For next year I'd suggest to possibly bring the bulbs in a bit earlier and/or leave the foliage on to dry out completely. This I think will help you remove the stems more fully and easily as the foliage will help to draw moisture out of the stem. At least this worked for me. This can be a slow process and most likely will take several weeks, not days.
Also I'm personally opposed to washing bulbs as I tried this with some Fritillaria imperialis for summer storage and the result wasn't good as I lost at least one bulb to rot. Also I kept the sand I stored them in very lightly moist. The following years I have just let them air dry a few days and then stored them directly in rather dry sand and not lost a single bulb to rot.
