"I have been working on a project recently" is clearly a perfectly grammatical sentence using the Perfect Continuous tense.

It is not clear whether "I have been being sick recently" is grammatically correct though. Of course, most English speakers would simply say "I have been sick lately", but does this necessarily have the same connotations as the other sentence? Perhaps "been" entails the continuous aspect, but I'm not certain.

Similarly, can I say "Starting tomorrow, I will have been being sick for two months now"?

By the way, these sentences read less awkward if you put the emphasis on the word "being", since this implicitly groups "being sick" together.

Edit: In lieu of Ham and Bacon's answer, one can replace every instance of "sick" with "confused" or some other adjective and the question remains the same.