You might get away with explicitly importing in your sub classes, but it might still not work, because of the order in which import() will be called, if it gets called at all, see below:

package Module::Monkey;
use base 'Module';
BEGIN {
  Module->import();
}
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Note that use base does NOT call import() on the superclass.

You're trying to use Module as a base class, and as a holder of constants to be exported, and as a "module loader" that loads all its subclasses. Life would be a lot simpler if you would drop at least one of these functions in another module, and I would recommend splitting them all out, which will make your code a lot saner:

# in Module/Loader.pm
package Module::Loader;
use Module;
use Module::Monkey;
use Module::Ape;
use Module::Constants;

# in Module.pm
package Module;
use Module::Constants;
sub some_inhereted_sub { };

# in Module/Constants.pm;
package Module::Constants;
use constant {
    MDB_INT        => 0,
    MDB_FLOAT    => 1,
    MDB_STRING    => 2,
};

# in Module/Ape;
package Module::Ape;
use Module::Constants;
use base Module;
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Etc.

Hope this helps,
Joost.