In good conferences (and I would certainly assume this to be the case in an A* conference) decisions are not made solely and directly by the PC Chairs, and definitely not only by looking at the scores. What usually happens is that after each assigned PC member submits their reviews, a discussion phase among the reviewers is triggered. In some conferences this happens in a long face-to-face meeting of the majority of the PC, in other conferences this is done offline through the reviewing system. All controversial papers (and a paper that gets two +2's and one -2 would certainly qualify as controversial) will be discussed - usually the PC chairs will ensure that both sides of the argument are heard, and then the reviewers and the PC chairs jointly arrive at a common decision (other PC members are also free to chime in, but in practice this only happens in face-to-face meetings in my experience).
Typically, there is little trace of this discussion in the reviews or notification letter, so it is difficult for you as an author to say if the two +2's did not have good arguments to fight for your paper, did not feel like they wanted to "champion" the paper despite their positive reviews, or were ultimately ignored because the reasoning of the -2 was more convincing. However, for a CORE A* conference I would assume that this discussion happened, that the PC chairs are quite aware of the difference in opinion regarding this paper, and made a conscious decision to reject the paper despite the positive reviews.
Hence, writing them is highly unlikely to change anything. For a large conference, PC chairs typically get multiple such requests each year, and I am not aware of even a single case in my community where the decision was retroactively changed. The standard answer for all questions on this site that fall into the category of "What can I do about this (from my perspective) unfair reject?" applies: move on, and try another conference and journal. Fundamentally, academic reviewing is a stochastic process, and even the best submissions can be rejected unfairly.
However, there still is something to learn from this experience. It's a difficult thing to do, but you should try to figure out why the -2 misunderstood your paper (and why apparently your paper did not give the +2's sufficient ammunition to fight for it effectively in the discussion phase):
- Was the contribution not clearly enough described? Note that "we had a strong contribution" isn't sufficient for an A* conference, it also needs to be very obvious that you have a strong contribution. A paper can, in fact, be rejected if the detractor makes a convincing case that your paper can easily be misunderstood (even if the champions manage to convince them that they indeed misunderstood).
- Did the paper get the wrong reviewers? One common source of unfair rejects is if the paper at first glance (e.g., from title and abstract) appeals to the wrong group of PC members (e.g., a paper looking like it has a strong theoretical contribution will primarily be reviewed by theory-minded folks - if it then turns out that the actual work is very practical, it will end very unfortunate, with reviewers misunderstanding the paper or simply not being interested enough to actually fight for the work).
- Was there some other issue that may have led to a normally tempered, sane, and competent person to vote for -2 (and, apparently, fight for a rejection - because the more normal case when +2 +2 -2 reviews come in is that the -2 gives in quickly and the paper is accepted)? Scan their review for this - maybe there is something that you would consider a minor threat that they consider backbreaking, or maybe you missed some related work that they consider crucial.
However, after all is said and done, your paper has in fact convinced two thirds of its reviewers - resubmitting a slightly improved version of the paper to a similar (or even the same) conference has in fact a very high chance of success.