As of Spring 19, I don't believe you can adjust
setReplyTo on Lightning emails via the UI. You can attempt to write a
before insert trigger on the
EmailMessage object and see if you can adjust the
replyTo property within the trigger.
If the trigger approach doesn't work, I'd suggest having your users setup email forwarding on their email accounts to your email handler. If you use G Suite, you can setup conditional forwarding, so that the forwarding filter logic could scan for a particular pattern that you use when emailing from the Lightning UI (like a standardized signature, etc), so that only relevant emails will be forwarded to your email handler