The Retired Officers Association is a tested model that could be used.
It is triservice and has volunteers who are elected to various offices. ROs are all in employment by the crown, same as regulars, and many wear uniform and perform identical functions to their regular colleagues.
The difference is they are outside the military food chain and can operate on behalf of their members regardless of which TLB their member works for.
If the regulars had a similar association you might have a Naval 4-ring who could be approached by an army officer who had a problem. You then have a disinterested party who can tackle the appropriate command.
Where this model may breakdown is the sheer size of the regular manpower pool and the potential for enough cases to become a full time job. That being the case the Association might have to pay for full-time staff too.
There is no case however for 'union meetings' 'do we agree to fight or shold we tippytoe away type.' 'Or but the S'nt Major told me to get me 'air cut, its an infringement of me human rights.'
The role of the association would be limited to ensuring that the current rules, as writ, are fair and properly applied. Where a properly written rule is patently unfair then the association could challenge it.