Is meaning-seeking behavior a biological optimization problem?
it seems to me that a meaningful existence requires
  the absence of cognitive dissonances, but I wonder what science would
  have to say about this, and if there is any empirical basis for
  claiming this.


The difficulty, as you've pointed out, is that there isn't agreement on the definition of "meaning" in this context.  Meaning can be interpreted by different people (and cultures) variously as: "accomplishment", "social good", "altruism", "realizing potential", "inner peace", "knowledge", and on and on.


  the argument that meaning-seeking behavior is a purely evolutionary
  phenomenon that serves only to maximize one's potential for
  reproduction doesn't appear sufficient.


Read about evolutionary explanations for altruistic behaviour for example to see how this can be addressed: "Every single instance of altruistic behavior need not always increase inclusive fitness; ... it would be beneficial to err on the side of caution and usually be altruistic even if in most cases there were no benefits."


  how far would we need to curb the definition so that humans would no
  longer qualify as the only meaning-seeking organisms?


"Social good" and "altruism" are examples of meaning-seeking behaviours commonly found in many animal species.


  Could modern experimental methods such as EEG, fMRI, gene expression
  profiling and epigenome sequencing be used to answer some of these
  questions at least to a degree?


Sure, why not.  For example: "Altruism, the experiment suggested, was not a superior moral faculty that suppresses basic selfish urges but rather was basic to the brain, hard-wired and pleasurable."


  I'd be happy to know at least if there are some research groups or
  labs that try to tackle such questions.


The area of psychology most likely to address such an abstract concept as meaning-seeking is probably Humanistic Psychology: "Human beings are intentional, aim at goals, are aware that they cause future events, and seek meaning, value, and creativity."

You can read about research done in this field - there isn't very much of it though - and also about self-actualization (another common interpretation of meaning-seeking in psychology).