Had he delayed, and not hurried to fight Midyan, could he have
prolonged his life indefinitely?
Prior to your question we must ask a more basic question: Would delaying prolong his life?
It sounds plausible that it would have, but it also sounds like that is not the point. Instead, the point seems to be that even though his death was the next major event, he still did it happily and without delay. His mindset is what we are focused on, not the timing.
But, nu, would his life have been extended? Well, let's see. After the war, Moshe gave a speech. Actually, he gave a very long speech, some rebuke, and some b'rochos before the rest. He was also command by Hashem to ascend the mountain to have a good look, and only then, finally lay down. Being so much would happen first, it is hard to link the war with his death.
But, nu, would his life have been extended? Well, had he tarried, and assuming that none of this tarrying would have affected the rest of what he and He did, it is reasonable to assume that his life would have been extended accordingly to allow everything to be done.
What if Moshe had tarried indefinitely? This is highly theoretical. I get the impression that would have caused rebuke from Hashem, and had he still tarried, he would have been deposed so Yehoshua could take over.
For comparison, see Shmos 4:24, where due to a bit of tarrying, Hashem sought to kill him. Though the variables are different here, tarrying does not seem to find good fortune.