The remote computer is domain joined, so Kerberos should be fine. I don't need to specify authentication type when using an administrative account. Is that not the case with non-admin users?
I am doing this as part of a script and would prefer not to put credentials in the script if I can avoid it.
It is on all computers with the Exchange management tools installed. If you look at the properties of the Exchange Shell shortcut, you will see that the Exchange Shell shortcut calls that script to load Exchange Cmdlets.
$session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri http://Exchangeserver_FQDN/powershell -Credential (Get-Credential)
Import-PSSession $session