While the hash is string-interpolated, is the hash-key inside an executed expression!

And ${test}_bar is not a valid expression.

Debugger demo:

  DB<28> sub EXPR { print "executed" }

  DB<29> "  $hash{ EXPR() }  "
executed

  DB<30> ${test}_bar
Bareword found where operator expected at ...
[download]

So multiple ways to do it are:

use strict;
use warnings;

my %test_hash = ("foo_bar" => "baz");
my $test = "foo";
my $key = "${test}_bar";

my $var1 = "aaa/$test_hash{$key}";
print("var1 = $var1\n");


my $var2 = "aaa/$test_hash{ $test . '_bar' }";    # concat operator
print("var2 = $var2\n");


my $var3 = "aaa/$test_hash{ qq(${test}_bar) }";   # qq() inside ""
print("var3 = $var3\n");


my $var4 = qq(aaa/$test_hash{ "${test}_bar" });   # "" inside qq()
print("var4 = $var4\n");
[download]

var1 = aaa/baz
var2 = aaa/baz
var3 = aaa/baz
var4 = aaa/baz
[download]

And since you introduced printf

my $var5 = sprintf 'aaa/%s', $test_hash{ "${test}_bar" };   # inside-o
+ut
print("var5 = $var5\n");
[download]

update

On a side note: Another good reason why perl-strings and here-docs are not a safe template syntax.