As you noted in the question, the accepted halacha is that one requires bread at the first two meals (Shulchan Aruch 274:4). At the third meal, one needs to eat bread as well; however, there are opinions that one fulfills his obligation with mezonot items (or even meat and fish, or possibly even fruit) (Shulchan Aruch 291:5).
Now, the opinion that allows one to follow kiddush with mezonot in order to fulfill the requirement of kiddush bemakom se'udah (that kiddush must take place in the context of a 'meal') is that of Magen Avraham 273:11.
However, I contend that even Magen Avraham believes that these mezonot would not count as his second shabbat meal. I base this on Magen Avraham 274:2 where he discusses someone who has no appetite to eat on Friday night. Magen Avraham writes that such a person should make kiddush on Friday night and eat/drink a little to fulfill kiddush bemakom se'udah, and then must eat three full meals on shabbat day.
So, even according to Magen Avraham, eating mezonot on Friday night does not count as his first meal. I contend the same is true regarding the second meal, as in your case.
However, it must be noted that some do hold that, bedi'avad one would fulfill his second meal with mezonot. Responsa Divrei David 86 (R. David Meldola, 18th century) writes that, although one has acted incorrectly in not eating bread at his second meal, he has fulfilled his obligation bedi'avad. He therefore rules that if one eats bread at a subsequent meal and forgets to add retzei in birkat hamazon he does not repeat, as this bread meal is considered the third shabbat meal, where forgetting retzei does not necessitate repetition (see Shulchan Aruch 188:8).