DATA is not set until the file is fully compiled. Change
BEGIN { load(); }
to just
load();

Whatever code is at the top level of the .pm or in BEGIN blocks will get executed before use or require returns, so it's still done "at the pt. when a program 'use's my module". Also, it will only get executed the first time use or require is called in a given interpreter, so it's safe to use the module from multiple files. For example:

# mod.pm
package mod;
sub load {
   print(<DATA>);
   print("loaded\n");
}
load();
1;
__DATA__
some data
[download]

# script.pl
BEGIN { print("Before 'use mod;'\n"); }
use mod;
BEGIN { print("After 'use mod;'\n"); }
use mod;
[download]

output:

Before 'use mod;'
some data
loaded
After 'use mod;'
[download]

Updated