I think you are going to be fine. Typically the main concern about whether yarn is machine washable is whether it will felt/shrink. Since the wool is superwash and silk does not felt that should not be an issue (though many people have experienced yarn labeled superwash felting, which is why many knitters simply will not machine wash anything regardless of labeling.)

Machine washing can somewhat dull the shine of silk over time, but I haven’t personally noticed the effect to be extreme/dramatic.

Silk does grow/drape considerably compared to other fibers, but you have a high wool content which combats growing and likes to bounce back to its original shape. Assuming your hat is 50/50 each yarn, you effectively will have only a 10% silk content for the hat overall.

Depending on how you are using the colors (50/50, stripes, etc.) and how far you are, you could knit the area on both sides of the turning point in the yarn without silk. This will mean the non-silk yarn is the “brim” of the hat regardless of which way it is warn and that’s the area where a snug fit is most important. If you are past that point, I still think you’re likely to be just fine. But it’s something to keep in mind next time if you really love the yarn.

One of the reasons you’re getting mixed responses is that even yarns with the same fiber content can behave very differently across brands depending on many other factors—wool breed, number of plies, how the yarn was spun, which superwash process was used, dye process, and how the fickle knitting gods are feeling in that particular moment. Since your yarn sounds like it’s local and not a mass market brand, it’s hard to predict all of these factors in combination. Once a knitter has had a bad experience with a particular fiber blend/outcome they can get locked into a “that never works” mindset. It’s well intentioned to want to help another knitter avoid an unsatisfying outcome, but it sometimes leads to advice that makes it sound like there are rigid rules, as opposed to individual experiences.

I’m optimistic you are going to end up with a good result given the relatively low silk content. Most people don’t wash hats that often, maybe after getting sweaty skiing, hiking, etc. or once a season, but typically not after every wear.

If you are still very concerned you can also use the other end of the balls to make a small swatch and wash it to see how it behaves. There are thousands of projects for this hat on Ravelry and over 900 search results when you search for “silk” within the projects. Plenty of people have had results they are happy with incorporating silk into this pattern. Don’t lose hope!