I’d say it’s better not to make assumptions about what the theater will provide, so bring your kit for at least the first few days. When you get familiar with the theater, and have a chance to see what they have onsite, then you can decide whether you want to keep your kit there or bring it home. I don’t think an SM with a kit would look dorky, but an SM without one may seem unprepared .
When you’re deciding how much to keep on hand, also think about the set up; is there an SM office, just a desk in the corner, or a storage closet to pull supplies from? How far is it from the rehearsal space? It may make sense to keep your kit with you so you don’t need to run down the hall every time you need something….and you can never have too many pencils, post-its, sharpies, hole punchers, or erasers, IMHO
I’ve worked in a mid-sized theater where using my mini-sewing kit was faster and easier than searching the costume shop for a needle and thread. I’ve also worked for a large theater where we needed four tape measures to tape out the set, but only had two on hand. Luckily, I was able to pull two more out of my kit.
Personally, I worry least about stocking first aid supplies, since Equity rules require the theater to have them. If the theater’s first aid kit isn’t in the rehearsal space, I’ll pull some of the basics from it to keep in a baggie wherever we are rehearsing.