tr doesn't do interpolation:

my $line = "ABABABAB";
my $limit = 3;
my $base = "A";

say "Yes 1" if $line =~ tr/$base/$base/ >= $limit;
say "Yes 2" if eval "\$line =~ tr/$base/$base/" >= $limit;
__END__
Yes 2
[download]

You need an eval to interpolate.