Here is another solution variant for you. The problem is easier if you are working with column numbers instead of names for the selection criteria. But I did that conversion for you. Column print order is same order as names in @desired_cols.

Code winds up with an ordered list of column positions to print and then does that for the data in the file by using list slice.

#!/usr/bin/perl -w
use strict;

my @desired_cols = qw(colname1 colname3); #order matters here

# reads first line to get actual column names

my $header_line = (<DATA>);
my @actual_cols = split(/\s+/,$header_line);

# get column number of the actual column names

my $pos =0;
my %col2_num = map {$_ => $pos++}@actual_cols;

# translate the desired col names into position numbers
                              
my @slice = map{$col2_num{$_}}@desired_cols;               

print join("\t",@desired_cols),"\n"; #header line
while (<DATA>)
{
   my @row = (split)[@slice];
   
   print join("\t",@row),"\n";       #each data row
}


=pod
Prints:
colname1    colname3
val1    val3
val11    val31
val12    val32
=cut

__DATA__
colname1 colname2 colname3
  val1     val2        val3
  val11    val21      val31
  val12     val22       val32
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