my $directory = "\.\\v706_7-17";  # Set directory to search
m|$directory\/(.*)|  # Variable way that didn't work
[download]

So $directory contains the string .\v706_7-17. As a regex pattern, it matches a string that contains any character followed by a vertical whitespace (\v) followed by "706_7-17".

You forgot to convert your literal text into a regex pattern. You can use quotemeta to do this, most easily via \Q\E:

m|\Q$directory\E\/(.*)|
[download]

Note that you probably want a ^ at the start of that pattern.