G'day leoberbert,

I think this does what you want:

#!/usr/bin/env perl -l

use strict;
use warnings;

my @files = qw{
    1020300000_XXXXXXXXX_20160707193000.TXT
    1020300000_XXXXXXXXX_20160707170000.TXT
    1020400000_XXXXXXXXX_20160707180000.TXT
    1020400000_XXXXXXXXX_20160707190000.TXT
};

my @sorted = map {
        $_->[0]
    }
    sort {
        $a->[1] <=> $b->[1] || $a->[3] <=> $b->[3]
    }
    map {
        [ $_ => split /[_.]/ ]
    } @files;

print join "\n", '@files:', @files;
print join "\n", '@sorted:', @sorted;
[download]

Output:

@files:
1020300000_XXXXXXXXX_20160707193000.TXT
1020300000_XXXXXXXXX_20160707170000.TXT
1020400000_XXXXXXXXX_20160707180000.TXT
1020400000_XXXXXXXXX_20160707190000.TXT
@sorted:
1020300000_XXXXXXXXX_20160707170000.TXT
1020300000_XXXXXXXXX_20160707193000.TXT
1020400000_XXXXXXXXX_20160707180000.TXT
1020400000_XXXXXXXXX_20160707190000.TXT
[download]

The construct I've used here is known as a Schwartzian Transform.

This sorts first on column1 and then on column3, both in ascending order; this should put your wanted file in $sorted[0]. If that's not what you want, modify the sort { ... } code to suit.