It's implied that Yuuma's side effect detected the fact that the generals weren't being honest with their attitude towards him:
"This is what Yuugo would have wanted."
"If we stop here Yuugo's efforts would have gone to waste."
These are all things they said to guilt trip and emotionally manipulate Yuuma into fighting for them. The scene before shows what their true thoughts and attitudes are — what they say behind Yuuma's back vs what they say to his face.
The lie is in how they put on a show as if they're acting on Yuugo and Yuuma's best interests, but in actuality they're only focused on their own.
If we want to be extremely technical about this:
"This is what Yuugo would have wanted" is a lie because Yuugo would have wanted Yuuma to make his own decisions.
"If we stop here Yuugo's efforts would have all gone to waste" is a lie because the results of his efforts still exist and it's impossible for his impact on the war to be erased that easily.
Ultimately perhaps Yuuma decided there is a kernel of truth there - if this nation falls, maybe his father's war efforts really would have come to nothing. However he does say his main reason is to "finish what they started together," and I believe there is a strong element of grieving in his decision.
'From there on, for a period of three years, Yuuma pursued the battle by substituting himself for his father.'
There is a sense that Yuuma continued because that was how he was coping with the loss of his father.
After all, every action he's taken since had been related to his father, from finishing what they started together in the war, to finding Border to find a way to trade back his life for his father's through reverting the Black Trigger. It's particularly telling in how he lost his purpose in life after finding out that's not possible, establishing the fact that he has no desires or wants or wishes outside of his father, to the point where his attachment to life is in question.
With that in mind, it's no surprise that he accepts any ready-made purpose delivered to his door, whether that be from the generals who wanted to make use of him for their war, or Osamu's request to help Chika go on the away mission.
Except there's a key difference this time. In contrast to the generals who weren't upfront with their intentions, with a request made for the sake of themselves, Osamu was entirely upfront with his intentions, and his request was made for the sake of Yuuma.