Girl, 5, fined £150 for running homemade lemonade stall

It was the girl's father who was fined, not the little girl.
The fine was dropped once the father pointed out the situation to the officer's superiors.
This was not a case of a child selling lemonade from their front yard etc. The father took the daughter and set up the stand in front of a high traffic area where people where thousands of people were attending a music festival.
Could the officer have handled it better - for example by simply asking them to pack up and leave, rather than fining them? Probably.
Should the father have realised there would be permits needed for setting up food/drink stands in front of a music festival? Given that he is a Professor of Business - definitely.
Should she be allowed to have a stall without a permit because she is a child? In my opinion, no. First, because even good causes such as charities need a permit. Second, because she was not operating on her own as a child but under the supervision and responsibility of her adult father. She physically should not and cannot be there operating a lemonade stall without her father, and thus why the responsibility (and fine) were ultimately his.
Had this been a case of a 5 year old play "lemonade stands" on a whim on her front yard, would the officer have acted differently? I suspect yes.
If all of the above information was reported in full without the sensationalism, would we be nearly as shocked or interested? No.
Would I be more likely to upvote you if you had in fact researched and detailed some of this in your post, or even given your own opinions instead of merely asking others to add value to your post with their research and comments? 100%