It appears you looked up the meaning of
相手 and found that it can mean
partner. It has that meaning sometimes, but in this case it is better to think of it as meaning
the person (or thing) that you are interacting with. So in the example sentence when it says
靴下を相手に, Scrooge is interacting/dealing with his socks.
With that in mind you could make the following literal translation:
He showed the appearance of a Laocoon in dealing with his socks.
Having said that, are you aware that you are reading a Japanese translation of a book that was originally written in English? The original English excerpt is the following:
“I don't know what to do!" cried Scrooge, laughing and crying in the
same breath; and making a perfect Laocoön of himself with his
As you can see in the original, it only uses "with his stockings", but I believe the Japanese uses
を相手に to emphasize there was more of a concerted effort (struggle) involved. Also, keep in mind this book is from 1843, so its is a Japanese translation of English from the 19th century...