Place of an adverb in the passive present perfect progressive

Solution 1:

Note that there are a few instances of "been being carefully" in Google Books, but no relevant occurrences of those three words in any other sequence.

Personally I have no real preference for carefully being placed before or after the verb - but in this context, "the verb" means the word "tickled", not the "helper" verb forms "been" and "being". Thus,

I have been being carefully tickled.
I have been being tickled carefully.

...are equally valid, and mean exactly the same thing. Here are a couple of real-world instances...

...has been being researched systematically...
...have been being carefully researched...

In short, put your adverb right next to (either immediately before or after) the "operative verb". Just don't splice it in between any pair of "auxiliary" verbs you happen to be using.

Solution 2:

Well, for one thing, it depends on the adverb. The placing of adverbs varies according to the type of adverb ('An A-Z of English Grammar and Usage' by Leech and others). Carefully is an adverb of manner, and such adverbs typically, but not invariably, occur at the end of a clause: ‘This path can be very slippery and walkers should therefore descend it carefully’.

However, perfect progressive passive constructions are very rare in English ('Cambridge Grammar of English' by Carter and McCarthy) and your example is such an improbable sentence that it would be hard to say where any adverb should or should not go at all.