(Our answer does not address the precautions that should be taken when some are eating dairy while others are /eating meat.[1])
The tosefta[2] states that kohanim and ordinary Jews can join together for zimun. The gemara[3] notes that this seems obvious and therefore posits that the tosefta relates to a case in which kohanim ate teruma (which is forbidden for a non-kohen to consume) while the others ate regular food. The reason that they can join together to form a group for zimun, says the gemara, is that although the ordinary Jews may not eat a kohen’s teruma, the kohen may eat their food. The Shulchan Aruch[4] states that if a kohen ate something non-kohanim may not eat and the others ate bread baked by a non-Jew that the kohen is careful not to eat (as it is forbidden according to certain halachic opinions[5]), they do not form a zimun together.
The Magen Avraham[6] raises the matter of dairy and meat, which you related correctly, and points out that the meat-eaters and the dairy-eater can join together because the one eating dairy can still eat meat. (Acharonim discuss how this reasoning fits in with the various cases and opinions regarding the necessary break between eating meat after dairy.[7]) One might have thought that this point is actually irrelevant; after all, the eating that creates the obligation of zimun is of bread, which is classically pareve. Why should it matter that the “side dishes” of the respective eaters are dairy and meat? The Magen Avraham is sensitive to this question and points out that the discussion applies to a case in which the bread is “soiled” with meat and dairy, respectively. There is a machloket about a case in which the food that one ate was off limits to others but bread that everyone may partake of is available,[8] but we assume that the matter is determined by what was eaten.[9]
The Magen Avraham is also the source for your assumption that since it is the dairy-eater who unites the individuals into a group for zimun, he should “make the beracha to exempt the others.”[10] Once they form a group for zimun, it is not entirely clear why it makes a difference who leads the zimun.[11] Indeed, some Acharonim do not cite this ruling,[12] and even some who do cite it view it as a minhag, not an absolute requirement.[13] It is also notable that the Magen Avraham cites the practice in a case in which only the mezamen does the bentching and exempts the others, whereas current practice is that all bentch separately and the mezamen only leads the beginning. In our day, then, the stakes regarding zimun are much lower. Therefore, it is not surprising that the Sha’ar HaTziyun[14] says that if one of the meat-eaters is a kohen, the more established halacha of giving respect to kohanim trumps the preference of the dairy-eater.
Regarding your case of four eating together, where there is a zimun of meat-eaters, the proper course of action probably depends on the Magen Avraham’s reasoning, which is not spelled out. If we prefer that the person who is most connected to everyone be the mezamen, it still makes sense to have the dairy-eater lead the zimun. If the Magen Avraham’s idea is to serve as a reminder that what the people ate can affect the viability of the zimun, then when the zimun does not depend on the dairy eater, it should not be necessary to choose him. However, one can raise counter arguments. In any event, as we have seen, the matter is not particularly significant, and it does not warrant worrying about which possibility is more likely, such that one can do as he prefers.