Oh man, what an insightful question. So, I don't know if this will help or not, but...
I think the web of ISB is frighteningly vast when the machinery is compelled to move accordingly. And when it comes to hunting down Jedi, it moves very well indeed. When we meet Bode, he was referred to Cal and his crew by Saw Gerrera... I think, between Bode committing to joining Cal's crew, and actually running the mission together to secure intel from Senator Sejan, there was a significant amount of time. And I think, in that time, an astonishingly large number of individuals who had met or run ops with Cal disappeared. The rendition process for enemies of the Empire would be terrifyingly efficient... and information extraction is a key skill of the ISB, from crude methods like the IT-0 Interrogation Droid to Force-Enhanced Interrogations. Slowly, a library of data was built up about Cal Kestis and his crew.
Cal doesn't really go to extreme lengths to hide his Jedi roots when he's running ops... all the ISB had to do was slowly, carefully infiltrate Saw Gerrera's resistance force and start identifying individuals who had crossed paths with Cal Kestis. Especially folks who had run with his team on missions. It would quickly become apparent what Cal's skills are, as I'm sure Saw would make certain to put them to good use. Cal's psychometry is a powerful skill for gathering intelligence... it would have become commonplace for Imperial items of value to run through Cal's hands, if only to see if there was valuable memories / events to be gleaned.
After Bode gives Cal the blaster, we still see him use dual-blasters all over the place. So... I can certainly see Bode finding a blaster, setting it aside somewhere, and then having it on-hand for that moment. It's also very possible that his time as a Jedi Knight working for Republic Intelligence gave him the training to use his Force Cloak ability to resist Force-Specific information gathering techniques. Cal's psychometry seems to often be tied to moments of powerful emotional or spiritual happenings... if he could just read anything and everything, he'd probably go crazy. There is, also, another side of this worth considering - Cal's psychological profile. Like many psychics from other stories, I'm sure Cal has developed a strong internal set of rules for when he uses his psychometry. Because if he just went around reading anything he could, I'm sure his teammates would see that as a real violation. Cal strives to embody the Jedi principles to the extreme... it may have been a safe bet for Bode that Cal would reject the chance to use his psychometry on the blaster because Bode is a friend, and he doesn't want to be eves-dropping on someone he respects and cares about.
Damn... I talk to much. Hahaha
Also, as for the laugh... honestly, I don't remember it! I'll have to look for your clip.
I will say this, though... for my personal take on it, Bode never actually intended to kill Cal. For all the crazy shit he does in that final battle, there were key moments where I left hints that Bode was still thinking rationally. When he's beating down Cal, and BD intervenes, Bode freezes for a moment, and then backhands BD out of the way. For a lot of fans, this was the moment where Bode became irredeemable. Which is funny, because for me, it was the moment he chose to spare BD. In full rage, Bode could have Force Crushed BD like a soda can. He could have grabbed BD and used him to bludgeon Cal to death. But he doesn't. He pauses for a moment, and then CHOOSES to swat BD out of the way. Why?
Bode knows that Cal is a better Jedi than he is. But more importantly... Bode has lost too much of himself to be a good dad to Kata anymore. He has fallen too far, and he knows it. When he asks Cal how far he'd go to protect Kata, Bode already knows that he's going to test him. And he knows that he will die in the process. Because that is the final test...
I think Bode, like many Jedi who were not enmeshed in Jedi Council politics, had reservations about the direction things were going. The insular, conservative nature of the Jedi Order led directly to its downfall. And there were many Jedi through those final years that had argued that things need to change... but nothing happened until it was too late. Bode truly believes that to survive in the world they are now living in, a Jedi must be willing to bend some of those rigid principles... and he pushes Cal to the breaking point. Making him so desperate and so angry, that he embraces his darker nature long enough to kill his adversary. Because, in Bode's eyes, that is exactly what Cal will need to do to keep Kata safe. And Bode will not rest until he is sure that Cal can go there.
This is super super deep BTS, but in the scene where Cal first meets Kata, Bode comes in and tells Kata that Cal is a friend. As I did so, I signed the letters O, K to Tajinae (Kata's actress). I had decided that they would have a silent form of communication because Bode would often have to say one thing out loud while needing Kata to know to do the opposite. So when Bode signs, "OK," to Kata, he's letter her know that Cal is safe to be around.
On one of the takes of Bode's final moment, Cameron shot me, and as I lay there, I reached out and signed, "O..." and then he shot me again. I was dying, and in that moment Bode silently tries to let Cal know, "It's okay." Because this is what Bode needed. And the only way he was going to be allowed to let Kata go, was through the actions of his best friend.