Could it be, that maybe, perhaps, the FBI took the shirt to Bledsoe to ensure that she had seen it before giving testimony and she screwed it up nevertheless?
As I said before, the FBI (on 12/4/63) had no way of knowing how the Warren Commission was going to proceed. According to Howard Willens (History Will Prove Us Right, page 35):
Hoover staked out a clear position from the outset: if the president had to have this commission, its function should be to receive the FBI’s report, review it, ask questions aimed at clarifying its findings, then endorse the report and disband.
That was apparently the mindset of the FBI at that point in time. The FBI had not yet even submitted its report to JFK. Yet, your fantasy has them plotting to establish something so that Bledsoe would testify a certain way? It is astonishing that anyone could actually believe such a scenario as you described it.
It's important to remember that less than 48 hours after the assassination Hoover had determined the outcome of the investigation.
The FBI were to find that Oswald was the sole assassin. No confederates. No accomplice.
"If the president had to have this commission, its function should be to receive the FBI’s report, review it, ask questions aimed at clarifying its findings, then endorse the report and disband."
Less than 48 hours after the assassination Hoover had determined the outcome of the FBI's report, that Oswald was the lone assassin. The Warren Commission was to view this report, ask questions to clarify it's findings, then endorse the report.
That is exactly what the Warren Commission did.
The result of the Commission had been determined before the Commission had even come into being.
And it is in this light that any FBI involvement in the investigation of JFK's assassination must be viewed.
This includes something as seemingly pointless as Bledsoe and the shirt.
Viewed through this lens it becomes obvious that the FBI visited Bledsoe to make sure she 'remembered' Oswald wearing the shirt that they already had fibre evidence for.
But it wasn't the shirt Oswald wore to work that day which demonstrates the FBI fibre evidence was as suspect as Bledsoe's memory and that it was all part of the charade to achieve the outcome of the investigation that Hoover had already determined.
This doesn't mean Oswald didn't do it.
It just means to take the FBI's report (and therefore the findings of the Warren Commission) with a pinch of salt.