Skip eval and hard-code tr///:

sub usingtr_d {
    my $a = $aa;
    my $b = $bb;
    # Delete ACGT first, assuming these are the
    # most common characters.
    abs($a =~ y/A//d - $b =~ y/A//d) +
    abs($a =~ y/C//d - $b =~ y/C//d) +
    abs($a =~ y/G//d - $b =~ y/G//d) +
    abs($a =~ y/T//d - $b =~ y/T//d) +
    abs($a =~ y/B// - $b =~ y/B//) +
    abs($a =~ y/D// - $b =~ y/D//) +
    abs($a =~ y/E// - $b =~ y/E//) +
    abs($a =~ y/F// - $b =~ y/F//) +
    abs($a =~ y/H// - $b =~ y/H//) +
    abs($a =~ y/I// - $b =~ y/I//) +
    abs($a =~ y/J// - $b =~ y/J//) +
    abs($a =~ y/K// - $b =~ y/K//) +
    abs($a =~ y/L// - $b =~ y/L//) +
    abs($a =~ y/M// - $b =~ y/M//) +
    abs($a =~ y/N// - $b =~ y/N//) +
    abs($a =~ y/O// - $b =~ y/O//) +
    abs($a =~ y/P// - $b =~ y/P//) +
    abs($a =~ y/Q// - $b =~ y/Q//) +
    abs($a =~ y/R// - $b =~ y/R//) +
    abs($a =~ y/S// - $b =~ y/S//) +
    abs($a =~ y/U// - $b =~ y/U//) +
    abs($a =~ y/V// - $b =~ y/V//) +
    abs($a =~ y/W// - $b =~ y/W//) +
    abs($a =~ y/X// - $b =~ y/X//) +
    abs($a =~ y/Y// - $b =~ y/Y//) +
    abs($a =~ y/Z// - $b =~ y/Z//);
}
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Benchmarks, including roboticus' robo_1, trizen's match, and choroba's tr_1e:

           Rate     tr   grep  tr_1e     sg  match robo_1   tr_d
tr        843/s     --   -18%   -56%   -75%   -80%   -92%   -99%
grep     1022/s    21%     --   -47%   -70%   -76%   -90%   -99%
tr_1e    1915/s   127%    87%     --   -44%   -55%   -81%   -98%
sg       3419/s   306%   235%    79%     --   -20%   -66%   -97%
match    4291/s   409%   320%   124%    25%     --   -57%   -96%
robo_1   9984/s  1085%   877%   421%   192%   133%     --   -90%
tr_d   100985/s 11881%  9784%  5175%  2853%  2254%   911%     --
[download]

Tested against these strings, which are ~80% ACGT:

my @alpha = ('A'..'Z', qw[ A C G T ] x 20);
my $aa = join '', map $alpha[rand @alpha], 1..100;
my $bb = join '', map $alpha[rand @alpha], 1..100;
[download]