The equivalent of

$invocant->$method_name( @args )
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is

($module.'::'.$function_name)->( @args )     # Perl 5.6+
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or

&{ $module.'::'.$function_name }( @args )
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---

use strict qw( refs ); forbids these (as it should), so you can use the following:

my $ref = do {
   no strict qw( refs );
   \&{ $module.'::'.$function_name }
};
$ref->( @args )
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Except it turns out that \&{...} is exempt from strict reference enforcement, so you can simply use

my $ref = \&{ $module.'::'.$function_name };
$ref->( @args )
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or even

( \&{ $module.'::'.$function_name } )->( @args )
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This trick to bypass strict reference enforcement is not documented.