So, you want to relay from the local host through a dedicated server. Lets call them the local host and the relay server respectively, but it would be useful to know the address of the relay: then I could inspect the certificate that it's presenting.
What's the address of the dedi? It's possible that your client is concerned about a host name mismatch, or a self-signed certificate.
To fix a host name mismatch, tell traccar to send email to the host name that's on the certificate that the relay host is presenting: likely xxx.dh.bytemark.co.uk, which may be self-signed. If it's not liking a self-signed certificate, then you could point the certificates in /etc/ssl to the certificates of a more suitable domain that has a letsencrypt certificate.
Or, you could circumvent all this by telling traccar to use your local Exim, and then have that deliver either to a mailbox on your relay server, or maybe configure the local Exim to route via the relay - that's the most difficult to set up, probably.