Enable Exchange Remote Powershell access for non-admin

Try the following:

$session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri http://Exchangeserver_FQDN/powershell -Credential (Get-Credential)

Import-PSSession $session

Probably is because you're not using the correct authentication:

Try it like this

Set-ExecutionPolicy RemoteSigned
$UserCredential = Get-Credential
$Session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri http://$ExchangeServerDns/PowerShell/ -Authentication Kerberos -Credential $UserCredential
Import-PSSession $Session -DisableNameChecking

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Source: https://docs.microsoft.com/en-us/powershell/exchange/exchange-server/connect-to-exchange-servers-using-remote-powershell?view=exchange-ps

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.

Resorted to providing the Credentials to the Cmdlet, but I still get an access denied error.

New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri "https://$ExchangeServerDns/PowerShell/" -Credential $ExOnPremCreds -Authentication Kerberos -ErrorAction Stop;

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Well, it's solved! :) glad you could figure it out :)

RemoteExchange.ps1 is that file available in all Exchange server or custom file you've created?

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.