I think the problem is simply that the [^\1]+ is greedy. Changing it to [^\1]+? seems to work:

$foo = qq{^snafu^|^foobar^\n};

$foo =~ m/\A(\W)  # \A instead of ^ and match first non-word
          [^\1]+? # Match everything that isn't in \1
          \1(\W)  # Match non-word following the 2nd \1
         /xms;

$text_qual = $1;
$field_sep = $2;

print "[$text_qual]\n";
print "[$field_sep]\n";
[download]

Output:

[^]
[|]
[download]

Update: Nope I think I'm at least partially wrong too. Greedy or not, if \1 is ^, then [^\1]+ should stop at the first ^.

Changing [^\1]+ to the hardcoded [^^]+ shows that greedyness doesn't matter. So it does appear to have something to do with \1 inside character classes.

Update 2: I think .*? instead of [^\1]+ might work, but it contains the dreaded dot-star. I assume the real solution uses some form of lookahead, but I've never been good with those.