I don't expect US tariffs to get added on for European shipments?
US tariffs apply to items imported into the US. We are discussing exports from the US here, right?
It's complicated, because it may be a re-export and tariffs can be deferred. You'll see this if you buy an imported part and a sec. 301 tariff is added to the line item. This is because a distributor doesn't have to pay tariffs on an entire shipment and can instead collect it from their customers as a separate fee, or tax. I'm sure if they turn around and ship deferred-tariff parts internationally they incur a documentation burden that makes small orders (like mine below, for qty 5) unrealistic. Don't think this is relevant to the OP, but I thought I'd just mention it for completeness...
BTW, ITAR is not a U.S. thing, it's an international treaty. It's just that some nations handle treaties as government or legislative guidelines, while in the U.S. when the Senate ratifies a treaty it does so in the form of adopting a
law. Which then applies to the government. If it's not law, i.e. the fallout of violating it is political rather than judicial, then it's much easier to adopt a "flexible" interpretation in the face of pressure from industry.