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$ perl -E '
    my $x = $ARGV[0] =~ s/(`[^`]+`)/$1/eegr =~ s/\R//gr;
    say $x;
' 'DMY_`date +%d%m%Y`_MY_`date +%m%Y`_Y_`date +%Y`.log'
DMY_21122019_MY_122019_Y_2019.log

$ perl -e '
    my $x = $ARGV[0];
...
    print "$x\n";
' 'DMY_`date +%d%m%Y`_MY_`date +%m%Y`_Y_`date +%Y`.log'
DMY_21122019_MY_122019_Y_2019.log

package Pm_11110263_FilenameGen;

...
}

1;

$ perl -E '
    use lib ".";
...
    say gen_filename($ARGV[0]);
' 'daily_~DATE DMY~_monthly_~DATE MY~_yearly_~DATE Y~.log'
daily_21122019_monthly_122019_yearly_2019.log

$ perl -E '
    use lib ".";
...
    say gen_filename($ARGV[0]);
' 'daily_~DATE DMY~_monthly_~DATE MY~_yearly_~DATE Y~_BAD_~RM ROOT~.lo
+g'
FATAL! 'RM ROOT' is invalid at Pm_11110263_FilenameGen.pm line 26.