Actually, "being sick" has the implication of 'throwing up', as in "During the entire boat trip, John was being sick," or "I was being sick all over the table". "I have been sick" means that you have been unwell, or ill, due to a disease.
They're both perfectly alright, they just mean differently.
Edit: In lieu of Rolfer's change of question:
Let's take the adjective "happy". The two examples are now "I have been happy" and "I have been being happy".
The first of the two sentences means that you were happy in times past. You might not necessarily be happy now, but you were happy before.
The second of the sentences however, meant that you have been doing some activity which occurs when you are happy, for example, Dancing, or jumping, or shouting, or singing, etc. "I have been being happy" denotes that you were doing an action associated with happiness. This can also be seen by "sick", or "angry" ('being angry' could mean stamping your foot, shouting, saying bad things, etc.).
Essentially, they mean the same thing, just stating it in a different way.
a. The Justice Secretary said prosecutors were allowed to join the event.
The first alternative should be used if no new participants are allowed after the event started at 9am, which might be the case here.
b. The Justice Secretary said prosecutors are allowed to join the event.
If it is possible for new participants to join the event subsequently (and later than 11am when the article was posted), it would be correct to use the latter.
Best Answer
This article from Education first [shortened and reformatted] explains the underlying grammar:
However, thoughts (recorded 'verbatim' or paraphrased, from either the character being referenced or the narrator, so grading into notions) do not demand inverted commas, so
Tomorrow would be the annual harvest fair – the biggest event of the year.
is acceptable.