I think that you're on the right track, Michael.... I've long believed that the umbrella was a reminder to JFK that he left the brigade with no aircover when he pulled the "Umbrella of air cover" for the Cubans who were trying to gain a foothold at Bay Of Pigs. Those Cuban's who were under the control of the CIA blamed JFK for their failure and capture by Castro's forces. It wasn't JFK's fault at all....The CIA was to blame for the failure...but they wouldn't accept the responsibility, and placed the blame on JFK. John Kennedy being the kind of man he was accepted the blame....( He felt that If it happened on his watch, he was responsible ) The red rings on the Windows of the TSBD were also there to remind JFK that he had betrayed the CIA trained Cuban exiles who were ashore at red beach , and needed the aircover that the CIA had promised them that JFK would provide.
I don't think it would have occurred to JFK in a million years that the pumping umbrella represented his refusal to provide air cover for the Bay of Pigs invasion force.
It is debatable that JFK refused to provide air cover. The story is much more complex than most history books paint it as being. It should be noted that the initial reports on the first air strike said it had been mostly successful, that it had destroyed almost all of Castro's air force--those reports were false. But, perhaps with these reports in mind, and concerned about plausible deniability and the uproar that the first raid had caused, President Kennedy probably felt it was both safe and prudent to cancel the second air attack. Nevertheless, when subsequent events proved that the first raid had not destroyed enough of Castro's air power,
Kennedy reauthorized a second air strike. It was scheduled for Sunday night, April 17. Unfortunately, there was a thick cloud cover that night, which made it impossible to carry out the raid. Moreover, after it became apparent that too many of Castro's planes had survived, JFK authorized the B-26s to bomb at will, and on the afternoon of the invasion one bombing raid destroyed an entire battalion of Castro's forces.