The whole point of setting up the EU subsidiary as a separate company that is incorporated in the EU and is managed and staffed by EU citizens is to avoid this.
The purpose of the CLOUD Act was to get at data that was stored outside the US but that was "in the custody, control, or possession of communications-service providers that are subject to the jurisdiction of the United States".
It arose from a situation where an email provider in the US used cloud storage services in several countries to store emails. They were asked for the email of a particular customer and said they did not have to provide it because they had happened to store that customer's mail at a non-US cloud provider.
What the CLOUD Act requires is that:
> A provider of electronic communication service or remote
computing service shall comply with the obligations of this chapter to
preserve, backup, or disclose the contents of a wire or electronic
communication and any record or other information pertaining to a
customer or subscriber within such provider's possession, custody, or
control, regardless of whether such communication, record, or other
information is located within or outside of the United States.
A company incorporated in the EU, even if it is owned by an entity in the US, is not subject to US jurisdiction and so that does not apply. The US owner is subject to US jurisdiction but the data of EU customers of the EU company is not in the US owner's possession, custody, or control.