No mention about Search Head cluster support in the doc from Crowdstrike. If you have a test environment you can test it. But i'm sure it's working on search head cluster, i have many apps in production with no mention for shc and it works perfectly.
Try to push the app with your deployer to the shc member (be careful to remove the inputs.conf file on the Search heads)
Thanks DenM. I appreciate the reply. We have a fully managed Splunk Cloud SHC -- we can't install such apps on our own. That is, the interface we have will indicate whether it is deemed SHC compatible or not. If not, we can't install it ourselves. And if we ask Splunk and vetting suggests it's not SHC compatible, they won't install it without remediation.