Currently it seems hard for us to further troubleshooting because it didn’t occur to all the emails. Did you find any obvious differences between these emails and normal emails?
“We use a connector to send the emails back to our on premise server to apply a signature.” Did you use the connector created by HCW? And how do you apply the signature? You could re-run HCW and only use the connector created by HCW, then check if the issue persist.
The network traffic communication between the Exchange Online and Exchange on-premises via TLS, as far as I know if the on-premises firewall which may affect the traffic and would cause this problem. So if possible and convenient, please try temporarily disable the on-premises firewall on a non-work time to see if it helps, thanks.