OK if you don't need those constants at all ...

Instead of passing the exported constants each time with ->process

>

    my $tt = Template->new(
        {
            INCLUDE_PATH => $BASE_DIR . '/root/src/',
            INTERPOLATE  => 1,
        }
    ) || die "$Template::ERROR\n";
    $tt->process( $email_args{text_template},
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you could make your config module export the initialized TT object

package Sam::Constants;
use Template;

our $tt = Template->new(                              # export me
        {
            INCLUDE_PATH => $BASE_DIR . '/root/src/',
            INTERPOLATE  => 1,
            CONSTANTS => { ... },
            VARIABLES => { ... },
        }
    ) || die "$Template::ERROR\n";
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(see also Re: Import Constants From File to Templates?)

And if things like $BASE_DIR need to be flexible, provide an import() function.

use Sam::Constants base_dir =>  "...";

edit

and if you have to manage different constructor setups, export functions which do the different construction and return the specialized $tt object.

update

package Sam::Constants;
my @defaults_all = (... );
sub foo { return Template->new( @_ , @defaults_all, DEFAULTS_foo ) };
sub bar { return Template->new( @_ , @defaults_all, DEFAULTS_bar ) };

... # other file
use Sam::Constants qw/foo/;
my $tt = foo(LOCAL_CFG);
$tt->process();
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