Don't know what you know, so this may be old news.
The machine name, and the domain name really have no effect on anything
outside your LAN. Now if it is made up of common 4 letter impolite words,
that could be an issue, but if it just says left.domain.lan when it should
say right.domain.lan, no biggy.
If it says left.domainname.com, that is the domain name, not the machine
name that is wrong, but the same answer applies, unfortunately.
The choice of pain is really yours, some pain now to start over, or pain
forever watching the "wrong" name.