Not sure how you would do this with an internal email system, but I do it with the gmail platform that has a business domain attached. All I've done is set up an email account, e.g.
support@mydomain.com and set that to forward emails to the gotoassist service desk email address. The customer's email address is shown.
I think it has something to do with the header that's sent. Your internal email system may be doing it differently.
Just posting to show it 'can' work.