Update: I read the question better & turned down the flame-o-meter on this comment...

This code might get you in trouble if there are newlines in the file. If that is the case then...

It puts "filename.txt" as every other line in the file. The while loop is what gets you in trouble here. Perhaps if you do a undef $/ somewhere in there.

Or, even better, start with your version and test for the first line.

perl -pi -e '$ARGV =~ s/\.txt//; $_=">$ARGV\n$_\n if($.==1)"' *.txt
[download]

This adds three things - it removes the .txt from the filename first. Second, it adds the initial '>' character. Finally, it only rewrites the first line. ($. is line number).

Not the most elegant, but it works.