We don’t use computer name at all as we have FQDN in Checkmk (We have multiple domains as well)
So we rely on the TCP/IP settings that I get in PS, like below. Hosts are added to Checkmk using automation and from our CMDB where they are always in lowercase so not a big problem, at least for us.
$Hostname = (gpv HKLM:\SYSTEM\CurrentControlSet\services\Tcpip\Parameters HostName, Domain) -join '.'