For most email service providers, having a valid DKIM record that lets the service know that the domain is authenticated to send emails is really important. additionally most transactional email services require a DKIM record before mail can be sent. We use Campaign Monitor to process transactional email and this is also the case with them.
I was hoping you would be able to provide a DNS record that authenticates Linnworks.net for originating email. It's a very simple process, you only need to setup a DKIM record and provide 3 things to customers:
1.) The selector i.e. transactionalemail._domainkey.linnworks.net.
2.) The Public Key
3.) The private key
Each customer would then be able to use the single DKIM record to authenticate Linnworks.net as the originating domain.