I entered a SolarWinds support ticket and got this resolved. For anyone else wanting to customize their Email subject lines without breaking reply-note functionality, the trick is to have "Ticket 999" together somewhere in there. The actual code is "{lbl.ticket} <var1>"
So my CustomLabels.properties file now looks like this:
email.msg.updated.subject.client = EngHD: <var4> [{lbl.ticket} <var1> <var2> (Updated)]
email.msg.updated.subject.client.closed = EngHD: <var4> [{lbl.ticket} <var1> <var2>]
email.msg.newTicket.subject.client = EngHD: <var4> [{lbl.ticket} <var1> <var2>]
email.msg.approval.subject.client = EngHD Approval Request: <var4> [{lbl.ticket} <var1> <var2>]
Which yields subject lines like:
EngHD: More email reply / note testing: Changed the su... [Ticket 587 Closed]
Instead of
Ticket 587 Open (Updated) --> More email reply / note testing: Changed the su...
Which is much more useful to us.