Backquotes interpolate variables the same way as double quotes. Backslash the dollar signs.

BTW: Why do you run perl from perl? You can do the work inside the script, as well (untested):

open my $IN,  '<', 'text.txt' or die $!;
open my $OUT, '>', 'text.new' or die $!;
while (<$IN>) {
    s/^(.{10})(.{382})(.*)$/$1^$2^$3/;
    print $OUT, $_;
}
close $OUT or die $!;
rename 'text.new', 'text.txt';
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