Darn it! I figured it out on my own just after posting my question. The solution I used was just to ignore the changing elements and use wildcards at the end. Specifically, I used the

my @tabrows = $p->look_down( _tag => 'a', 
href => qr{^\Qcheckme.php?player=\E .* $}x);
[download]

and that seemed to fix my problem. Onto more debugging! Sorry to waste the bandwidth and user time.