If I'm understanding the problem correctly, there is no problem.

Are you saying that you need INTERPOLATE => 1 to get [% INCLUDE "$screen" % ] to work?

This works ok for me:

use Template;
my $TMPL = Template->new();
my $res;
$TMPL->process( \*DATA, {
                          screen => qw(screen1 screen2)[rand 2]
                        }, \$res );
print "$res\n";
__END__
[% INCLUDE "$screen" %]

[% BLOCK screen1 %]
This is screen #1.
[% END %]

[% BLOCK screen2 %]
This is screen #2.
[% END %]
[download]

From the docs:

The INTERPOLATE flag, when set to any true value will cause variable references in plain text (i.e. not surrounded by START_TAG and END_TAG) to be recognised and interpolated accordingly.

and you don't need to do that. In fact you really don't want to do that..