... I'm dealing with large hash structures and big input data ... I'm usually don't even want to supress the error (this usually means the input data is wrong and I need to deal with it) ...

The point raised by haukex here is critical: you're really dealing with input data validation, not a way to flag the use of an undef value that might be present if your input data was invalid. In other words, don't bother trying to lock the barn door after the horse has bolted. The ideal time to deal with bad data is when you get it, not after it's drifted downstream and had a chance to gum up other processes.

The only way I can think of to validate a data structure is to write a validation function specific to that structure and invoke it as soon as possible after the structure is captured:
    my %bigdata = capture_big_input_data($inputstream);
    validate_bigdata(\%bigdata);
Obviously, the bigger your data, the bigger your validation job, but that's life. (You may even decide that certain missing data fields can, in fact, be fixed up with a zero, an empty string or whatever; if so, the  validate_bigdata() function is the ideal place to do this patching.)

The other variation on this theme is when you read your input data on a line-by-line or block-by-block basis. In this case, you must figure out a way to validate each line or block as it is read — at least, that's what I would do. Anyway, you get the idea...