Why was Lasky on board the Nimitz?

I thought it was to rescue the dog.

I believe the following might be at least a few reasons Mr. Tideman would want to send Mr. Lasky back in time if he believed it was necessary to maintain the timeline continuity.

1. It was Mr. Lasky who was the first to observe that the recon photos taken by the overflight of Pearl Harbor matched those of the historical photos in Commander Owen's cabin.That along with the other information helped convince the captain played by Kirk Douglas that the Nimitz had indeed traveled back in time.

2. Mr. Lasky also highlighted the fact that it was Senator Chapman whom the Nimitz crew rescued from the Zero attack. Senator Chapman was the vice chairman of the armed services defense committee. History recorded him as being missing and presumed dead. Mr. Lasky pointed out that Senator Chapman was a strong advocate of the theory that Japan could be an aggressor against the U.S. in the Pacific.

While events may have continued mostly the same without these actions, Mr. Tideman may not have been sure. Thus, wanted to take every opportunity to preserve the timeline to the best he could.

That leads to a possible reason 3 which may be more the most important in terms of continuity.

3. By Mr. Tideman requiring to have Mr. Lasky be on board, it delayed the launch of the Nimitz by two days. If it had launched earlier perhaps Mr. Tideman thought the Nimitz would not get caught in the anomaly. Thus the time loop would collapse.

As far as we know there is no time travel. But time travel and the resulting paradoxes have been a trope in science fiction for a long time.

It seems consistent with the movie that Tideman deliberately attempted to create a stable time loop. Tideman remembered Lasky being on the Nimitz, so he made sure that the initial situation was set up exactly as he remembered it because he had no idea what would happen if he didn't do that.