I watched the Columbo episode “The Bye-Bye Sky High IQ Murder Case” for the second time, and have come to believe that Columbo knew about the umbrella in the chimney and had seen it before it was removed by Brandt. He thus had evidence from the beginning and was watching for who would remove it.
Why do I think this? When Brandt goes into the room, escorted by an officer, to meet Columbo for the first time, he is left alone, but first we see a man exit the room with a shop vacuum shortly after he enters. Later, when Danziger presents his suicide theory, he mentions the possibility of the gun being pulled into the chimney. Columbo smiles, says “You’re not gonna believe this sir, but I, uh…” and gives a look like “ironically, you’re sort of right”, then shakes his head, walks to the chimney, and says that he “just happened to look up there” with “… all the lab boys […] vacuuming the rug […] said that there was powder carbon around”. To me, that points back to the man with the vacuum leaving the room when Brandt came in, and thus timing Columbo’s look into the chimney to a point where the umbrella was still in it.
If this is the case, it makes Columbo’s interest in Brandt’s umbrella in the park more than just his noticing of oddities, and makes his “honest mistake” of taking the wrong umbrella more clearly with reason, fruit of the forbidden tree not withstanding.
It is probable if they found the gun that they dusted it for prints at that time. They presumably didn’t have Brandt’s prints on file, and the lack of attempt to acquire them along with the lack of attempt to get a warrant for the umbrella suggest that Columbo was certain that he could get a confession and didn’t want to tip his hand. And of course Brandt did. And if Brandt denied it when they got downtown, there’d be the fingerprints taken from the gun earlier to compare to the now fingerprinted Brandt. But he seemed pretty resigned.