The very short answer is that Harewood is mortally afraid of Harrison. He seems to be aware of Harewood's true identity, and hence would be aware of his personal capabilities, monstrous intellect and capacity for unreasoned violence.
Given what he had managed to learn about the individual who had
provided both vial and ring, he knew he should have expected nothing
If he fails to keep up his end of the bargain, the clear implication is that Harrison will track him down with consumnate ease, killing him and his family wherever they try to run and hide. Note that (at least according to the novel) he does briefly toy with the idea of not going through with it, but then he sees Harrison in the street and decides against it.
Across the street stood the man with whom he had struck the bargain. The man who had somehow, in defiance of everything Harewood and his wife had been told, saved their daughter from a slow, certain death. He was watching. Quietly, calmly, without the slightest sign or suggestion of concern.
Harewood turned and entered the building. The arcade was old, perhaps
eighteenth century. He was resigned now. At peace with himself. Renege
on the agreement, and the miracle for which he was about to trade
everything might evaporate, its promise never to be fulfilled. That
was what he had been told, anyway, and he had no choice but to follow
through on the last of his instructions. That was the warning Harewood
had been given.
There's also a certain element of personal pride at play here. He's made a bargain (albeit with the devil) and he's going to see it through.
Can’t back out now, he thought. It didn’t matter. He had long since
resolved to see the matter through to its end. It was not about him,
Star Trek Into Darkness - Official Novelisation