Archer knew what he was doing
Trip: A poop question, sir? Can't I talk about the warp reactor or the transporter?
Archer: It's a perfectly valid question.
In season one of ENT's "Breaking the Ice", the crew of the NX-01 records a video for Ms. Malvin's class of fourth graders back on Earth, and one of the questions is what happens when someone aboard the ship flushes the toilet, which Archer throws to Trip, and Trip is concerned the kids are going to think he's the ship's sanitation engineer.
Now, because Archer is an awkward goober in this episode, it seems like he's reading the questions off the cuff, with no one other than the possible exception of Ms. Malvin herself, having first vetted them. However, at the start of the recording, Archer announces that he's the one who selected the questions so he knew going in that the, as Trip puts it, "poop question," would be in there. Also before the recording Archer told Trip that he needed to there to participate as opposed to dealing with the large amount of work that was on his plate. What's more, we know from the start of the episode that Trip's nephew is one of the fourth grade students in Ms. Malvin's class.
So, my theory is that Archer made the intentional choice to have Trip answer a question he knew the engineer would find to be embarrassing, specifically to diminish him in the eyes of his nephew. Why would he do this extremely petty thing to someone who is ostensibly his friend and most loyal officer? Probably because he sucks and wanted to put Trip in his place for some imagined slight, just like he later has an outburst with the Vulcan captain whom he invited to dinner.