What is the technical name for the public display of personal traits that are unproductive or unhealthy as an attempt to make oneself seem more likeable or relatable? (e.g procrastination)
This may not fit exactly the scenario described, but some form of strategic affect display or 'appraisal theory', these relate to social theories of (& displays of) emotion, as oppose personality traits as such, but may be relevant nonetheless.
So for example, you might want to come across as being more aggressive & assertive, or alternately more compassionate & caring, in any given scenario, because you feel that is the appropriate thing to do, or you may gain some advantage from that - even though you may feel otherwise indifferent about whatever it is.
Having said that, the way the question is phrased - whether something is 'productive' or 'unproductive' in a social sense, as oppose to a personal level, is open to interpretation, i.e. if it makes you more 'likeable' then it is socially productive (at least in the short term), regardless of whether it it something that is generally regarded as being a negative trait.
Hope that's relevant.
There is a related concept to this in evolutionary biology called the Handicap Principle or Zahavi Principle . It originally tried to explain why certain species evolved traits that are clearly not adaptive, but seemed to be favoured by sexual selection. For instance, the colourful plumage of the male peacock makes the male a target for predation, but may signal superior fitness to females. The logic behind is could be that the male is showing off that he can afford these non-adaptive nornaments and still get away with it. The kind of behaviour that you are describing may serve a similar purpose.
 Zahavi, Amotz (1975). "Mate selection—a selection for a handicap". Journal of Theoretical Biology 53 (1): 205–214. doi:10.1016/0022-5193(75)90111-3