I actually like Klawans's thesis, and use a lot of his insights in my dissertation. Interestingly enough, however, the idea that the ancient sacrificial cult was a symbolic system did not begin with Mary Douglas, Jacob Milgrom, and Jonathan Klawans, but St. Augustine (and perhaps earlier), who suggests a different unifying principle: love.
If in times gone by our ancestors offered other sacrifices to God, in the shape of animal victims (sacrifices which the people of God now read about, but do not perform) we are to understand that the significance of those acts was precisely the same as that of those now performed amongst us--the intention of which is that we may cleave to God and seek the good of our neighbor for the same end. Thus the visible sacrifice is the sacrament, the sacred sign, of the invisible sacrifice. (Augustine, City of God 10.5; trans. Bettenson)
Augustine is attempting to create an equivalence of ancient sacrifices with current practices through INTENTION. The intention is to cleave to God--perhaps close to attracting and maintaining the divine presence, except going the opposite direction: the "sacrificer" here ascends to cleave to God rather than having God descend upon the temple--and help one's neighbor do so also. In the process, we also get Augustine's famous definition of sacrament as a visible sacred sign (signum) that points to (signifies) an invisible reality (res). The invisibility of which he speaks, however, again comes back to intent--he culls up passages from Psalms and prophets (Ps. 16:2; 51:12, 18f; Mic. 6:6f; and Hos. 6:6) to focus on the heart. Yet we still have not really gotten to the organizing principle that keeps all of these practices of sacrifice together--whether the animal sacrifices in the temple, or the intention of the heart, etc. It, as noted above, has to do with love:
The instructions about the multifarious sacrifices in the service of the Tabernacle or the Temple are recorded in Scripture as divine commands. We see now that they are to be interpreted as symbolizing the love of God and the love of one's neighbor. For "on these two commands the whole Law depends, and the Prophets." (ibid.)
The quotation is Matt. 22:40. So, I think Augustine would say that the organizing principle of the sacrificial system is love of God and love of neighbor (Jesus' organizing principle for the entire Torah), while, in fact, he might say that the function of sacrifice is union with God. This, in fact, is the flip side of Klawans's function of maintaining the divine presence. Perhaps, too, for Augustine to love god and neighbor would also be an act of imitatio dei, given his particular view of the inner-workings of the Trinity (see De Trinitate) based upon mutual love.