Issue with email forwarding to Verizon cell as text with Office 365

For years our help desk has been alerting us to high priority tickets by sending an email to a distribution group in Exchange and then that's been forwarded to an email address associated with our cell phone carrier.  We've discovered recently that changes Verizon has made with their Spam filtering is causing this to be blocked so we're not getting text messages from them.

After much testing Verizon has confirmed that we suspected. They have a couple of possible solution, but we're not sure they're even possible. One of them is to reconfigure Office 365 in this situation to remove any of the forwarding headers so that it makes a new clean email to the cell phone email address. Before moving to Office 365, Exchange would accept the email from the help desk to the group and then send it out from the group to the group member, which is a contact with the cell phone email address in it.  What Office 365 does is to keep the original sender and then forward it out. We originally had a problem with the relay aspects of this, but I fixed that in our email filtering system by allowing that as a relay. I tested this back in January and it worked fine. So Verizon has made a change since then that is causing this. Since this change is something global for all customers to help hold down on phone spam, they're not going to change anything on their side.

So, is what they're suggesting even possible?  Can I have Office 365 basically make a new email without all of the forwarding header info to send to the cell phone email address?