I refer you to my earlier post here: RP578's post about the Black Watch request
The Black Watch line came from Dean Rusk in a heated off the cuff remark to Times reporter Louis Heren, that's all it ever was. Ellis in her book, attributes the bagpipe band request to Wilson's written recollection of a conversation with Johnson, although I've not seen the original source. Again, likely an off the cuff remark with no bearing on anything in reality.
I suppose that "Heated off the cuff remark" etc was made during the interview where Rusk went on to say, "When the Russians invade Sussex, don't expect us to come and help you." ? LBJ had charged Rusk and his State Department with rustling up international military support for the Vietnam adventure, and personally asked Harold Wilson to provide even a token force, for political and psychological reasons. According to an article in the New York Times a couple of years back, Robert McNamara (US Secretary of Defense throughout much of the Vietnam war) is on record as saying that he would pay a billion dollars for a British brigade.
No doubt the reason for specific mention of the Black Watch by either LBJ or Rusk was that awareness of the regiment in the US was due to the fact that their Pipes, Band and Drums were on tour in the US at the time of JFK's assassination. They had, in fact, given a performance on the White House Lawn just a week before Dealey Plaza. JFK and family definitely attended this performance, and his VP LBJ probably did. Not long after, the pipers marched in the procession at the funeral in Washington.
When considering the possibility of clandestine British involvement in Vietnam, it's worth considering the temper of the times in UK. There was much restrained chatter in some quarters about the possibility of a military coup in UK, due to suspicion of the motives and allegiances of Labour politicians, amongst others. It is now common knowledge that Wilson was being monitored by MI5. Most of the WW2 spooks and special forces founders/leaders were still alive and active, and the SAS was virtually unknown, and in many ways a law unto itself. As late as the early 1990s I heard a senior ex-SAS officer say, in effect, "That embassy siege was the worst thing that ever happened to the regiment, because it brought us into the public eye. We need to form another regiment nobody has ever heard of, and get back to having some fun."
That there was the will in many British quarters, to become involved in Vietnam can't really be denied. I felt personally ashamed at the time that we were not, at least, standing by our ANZAC friends.
That there was a clandestine opportunity is obvious - US SF were more than capable of providing a haven for a few Brits. Even General Harold Johnson, Chief of Staff of the US Army, stated that his own SF were "fugitives from responsibility who found a haven where their activities were not scrutinised too carefully."