A quick test may be to try swapping the #2 injector and spark plug with one of the others to see if the error code changes.

Also take a look at post #3 at this link.

I believe it was P0302.

What are the codes that you read from the obd port (Pxxxx)?

Now that it is suggested, it does seem to happen more frequently post fillup. I would add to my symptom description, that once the warning lught does engage, there is a different sound to the engine......a mild sputter and acceleration is sluggish. I read that the code 302 refers to a misfire in cylunder 2. I wonder if the computer shuts down cyl2 as it thinks something is wrong. Coil firing or injector? Because after the venting process, the xar runs great and sounds normal again.

I can't help specifically, but to me, the 30-second tank venting relieving the problem suggests that the tank isn't vented enough. Perhaps whatever allows air into the tank to replace the fuel pumped out is partially blocked. When you drive it hard, the pump sucks fuel out and finds increasing resistance or a sensor reads lowered tank pressure. 
Is it more pronounced with a full tank?

I  have  a 2013  Cooper  (non-S)  with  just  over  77,000km  on  it  and  recently  have  been experiencing  a problem  with  a  check  engine  warning.  Bought  certified  from  a  local  Mini/BMW  dealer  just over  a year  ago.  When  the  vehicle  is  driven  lightly  or  with  medium vigor,  no  issues.  Push  it  a bit  harder,  say  3/4  throttle  or  WOT  under  load,  the  engine  will  produce  a stammer  briefly  and  then  the  engine  warning  lamp  on  steering  gauge  shows  pic  of  engine  shape  with  diagonal  half  filled.    Venting  the  gas  cap  for  30  seconds  seems  to  remedy -  so  purchased  and  installed  new  factory  gas  cap  thinking a  leaky  cap  may be issue but  issue  still  occurs  under  hard  load.  Used  a code reader  and  got  a generic  302  code  which  discusses  details  regarding  misfiring.  There  was  a subcode  but  will  have  to  redo  to  post  that  detail.  Anyone  else  experience  similar  and  have  resolution?   Will  be  doing  plugs  (dealer  purchased)  shortly;  could  it  be  related  to  coils,  wires,  injectors, filters, etc?   I  am  leaning  the  direction  of  pressurization  of  fuel  system  though  as  gas  cap  thing  clears  it  up  each  time.  Any  guidance  will  be  greatly  appreciated.