How to add an external email domain to SWSD

We have recently acquired another company and we want to accept their domain emails incoming to our established configured support email address to be able to create incidents in our SWSD portal and in turn be able to have our incident responses transmit back to them as well

We have tried to configure the Allowed Addresses list with their domain specified in SWSD Email Settings but have found the result of this is that this interrupts our internal flow of incidents and not all of them are populating in the portal with this turned on. 

While the external domain incidents are created with this option turned on and configured for their external domain, we are not able to transmit the responses back to them.  

We have tried to address this from our 365 Exchange\Azure side by adjusting mail flow rules but this also has no positive effect and we still are not able to successfully achieve the end result.  We are missing something it appears on the SWSD side.

At this point to restore internal operations to normal, we have turned off the allowance for the external address and have implemented a workaround for them to email to a samanage email alias address.  But this communication is also one directional: to us but not back to them. We are communicating incident changes and updates directly to them via direct email. 

Any suggestions on how to get this properly configured to work seamlessly?  I appreciate any feedback and words of experience.  Thank you.