You might think of silkworm silk proteins as being composed of a single piece of Velcro attached to a thread. The thread entangles with other pieces of Velcro, binding the proteins together.
Spidroin is composed of repetitive sequences of Velcro, string, Velcro, string. Silkworm silk has proteins consisting of one big piece of Velcro and one string. Spidroin is many small pieces of Velcro attached with many pieces of string.
KBLB found a way to toughen fibroin. We don't know how. But all of that is beside my point. I brought up different properties of silkworm silk that don't depend on toughness. Toughness is measured by pulling on a fiber until it breaks. Washing machines don't put much extensive stress on clothing. They fold clothes over and over. A tough fiber doesn't mean it resists folding. Silkworm silk is reasonably tough, but washers can ruin silk because the fibers are damaged when they are folded. Will a washing machine ruin Bam-1 Alpha by repetitive folding? Not sure, but it looks like a reasonable possibility.
Can one throw a garment made by spider silk into the washing machine? I doubt there is much evidence, one way or the other. Certainly might be an issue.
To my knowledge, Thompson has never addressed the washability question of their silks. Which supports another statement I made: we don't know a lot about KBLB's silks.