There are two parts to your question: why would a pacifist sign/write such a letter, and why didn't Einstein work on the latter project? I'll address them each below.

\1. The Einstein-Szilard letter in no place advocates building a nuclear weapon or using it. It advocates the government centralizing some of the research into whether nuclear weapons are feasible. That's it. It is very modest in its requests when you look at it on its own terms, not in terms of the work the US government eventually did on the issue. (While 1939 was the entry of the US government into fission research, the real work to build a bomb did not start until mid-1941. I don't think the letter is as important as many people treat it to be, to be honest.) Here's the part on recommendations (which almost nobody seems to read — they get stuck on the part where they try to set the scene of why this is worth doing):

In view of the situation you may think it desirable to have more permanent contact maintained between the Administration and the group of physicists working on chain reactions in America. One possible way of achieving this might be for you to entrust with this task a person who has your confidence and who could perhaps serve in an inofficial capacity. His task might comprise the following:

a) to approach Government Departments, keep them informed of the further development, and put forward recommendations for Government action, giving particular attention to the problem of securing a supply of uranium ore for the United States;

b) to speed up the experimental work, which is at present being carried on within the limits of the budgets of University laboratories, by providing funds, if such funds be required, through his contacts with private persons who are willing to make contributions for this cause, and perhaps also by obtaining the co-operation of industrial laboratories which have the necessary equipment.

So, again, this is not recommending making, much less using, a weapon. Were they so naive, to think that a weapon might not result from this work? No. With Szilard it seems clear that the threat of Nazis with a bomb was so overwhelming in his mind that a weapon was worth building, maybe even worth using, to stop them. But there were many who saw the need for a weapon as a hedge, a deterrent, something to have to threaten to use, but maybe not to use.

\2. Why didn't Einstein work on the project? He wasn't asked to work on it. In fact, he was deliberately excluded from working on it. The military security people did not want to issue him a security clearance, because of his strong pacifist, socialist, and (in their minds) subversive views. (Einstein supported such radical ideas as Civil Rights and equality irrespective of race, and that made him a subversive in the eyes of J. Edgar Hoover's FBI.)

If the scientists had really wanted him to work on the project, it is imaginable that they could have found a way to do it — they managed to override the security objections on several other important, key people (Oppenheimer himself is the obvious one, but Arthur Compton also had enough red flags in his security file that military security initially denied him a clearance). Vannevar Bush, the scientist-administrator who considered whether to bring Einstein in or not, decided against it after consulting, through a "cut-out" (another scientist, so Einstein did not quite know what he was working on), with Einstein on a problem related to the enrichment of uranium. Einstein's response was a very theoretical treatment that had little feel for the practicality of the problem. Bush concluded that while Einstein was obviously a very great physicist, the areas of his expertise and work (e.g. Relativity and Cosmology) were largely not very useful when it came to building an atomic bomb (which, on the theoretical physics side, is more about nuclear physics, particle physics, and things of that nature). So he declined to bring him in on the work.

Isaacson's Einstein: His Life and Universe, while not the best Einstein biography for a lot of reasons, is actually fairly good on the whole circumstances of the Einstein-Szilard letter. Fred Jerome's The Einstein File is definitive on his issues with the FBI. The description of Bush's considerations of Einstein come from correspondence in the Bush-Conant File microfilm collection. My comments on Compton, security forces, etc., come from my own research.