Use $1 instead.

While backrefs like \1 work in the substition part of a regex, it only works in string context. The $hm{KEY} is accesses an item in a hash. The KEY part can be a bareword or an expression. In an expression, \1 would be a “reference to a literal scalar with value 1” which would stringify as SCALAR(0x55776153ecb0), not a back-reference as in a string. Instead, we can access the value of captures in the regex with variables like $1.

But that requires us to capture a part of the regex. I would write it as:

s/(Cat)/$hm{$1}/;

As a rule of thumb, only use backrefs like \1 within a regex pattern. Everywhere else use capture variables like $1. If you use warnings, Perl will also tell you that \1 better written as $1, though it wouldn’t have detected the particular issue in your case as the \1 was still valid syntax, albeit with different meaning.