(This is rather constrained; you want the hearer to be able to pick something that speaks perfectly, as anomalous, given speech as the only medium.)
You specify that the AI is limited, but I can not see how to leverage that without a conversation (which is excluded)… except as below.
I like the idea of some anomaly in the speech (especially the breathing aspect), but other answers have that pretty much covered.
One possibility is to have an electrical or mechanical sound made by the machine. One obvious one is interference between the AI’s speakers and the telephone electricals. [I think I read that in another answer, but I did not find it in a quick search.]
The mechanical alternative is, for instance, that the AI “holds” the telephone handset in some way, and there is a mechanical noise such as a cooling fan hum, or a fast clicking caused by vibration, or some noise caused by the AI holding the phone way too tightly, or the hum of hydraulic pressure being maintained… that is conveyed far better through contact that through the air.
In another vein… I am assuming that the bomber is using the AI purely to avoid using their own voice. The other possibility is to build on this aspect, and have a bomber who actually has some defect themself. Perhaps they are from a different sub-culture, and have chosen an AI that, for instance, says “Hello!” in a way that immediately puts anyone from this city on edge, but sounds fine to the bomber.
Perhaps the bomber has tuned the AI‘s speech to sound good to the bomber, but [also] such that it sounds artificial to others.
Perhaps the bomber has poor eyesight, or is not familiar with the phones in some way… such that they position the AI holding the phone such that it blocks the microphone with a “body part” that has peculiar auditory characteristics, such that the person at the other end will notice.
Perhaps a tangential idea is acceptable, such as that the AI produces speech with two simultaneous “carrier” pitches (pitch of voice, in a human)… but the poor bomber is tone-deaf to the higher pitch.
Perhaps the bomber is slightly deaf, and has turned up the volume too high on the AI’s speech — noting that a human being speaking loudly sounds different from that.
I like the idea that the AI is a standard one that (for instance) was used in a huge telemarketing campaign recently, and everyone will recognise its voice instantly… but somehow the bomber missed out on this.
I like “John Dvorak”s answer here; the bomber might be listening along to the AI, but not hearing the other end, and the AI might be completely screwing up the conversation, while delivering perfectly, lines that are inappropriate.