I would think that the public statement on tanno alone could be an interesting legal matter. An employment relationship is based on mutual trust.
Being publicly suspended and having employees of the employer undertaking private briefings could be seen to have broken that trust and made it impossible to continue working there.
But other than 'no comment', the Club would find themselves in an impossible position on that basis when questioned on Poyet's post game comments. Effectively as soon as Gus commented post-Palace he had started the ball rolling and betrayed the Trust you refer to. I would feel that within the contract of any BHA employee, there would be a statement regarding acting in the best interests of the club at all times (ambassadorial or alike).
But if there is an issue the LMA would have commented, its not like they've been quiet in the past.
Tanno as you say is interesting, but the club may have saved themselves by saying there is no case to answer post investigation. I don't know.
Charlie as I say, I really don't know enough.