I like a lot of what's being suggested here. "Blood of the Covenant" makes a lot of sense, and suggests the literal bloodiness of circumcision (which is most often referred to simply as "the convenant" -- brit or bris).
The Jews very much thought of themselves as being a family of descent -- the House of Israel, Israel here being a person, Jacob, not a geographic area -- more than a religion or nation in the modern sense. A family you could join, thus not strictly genealogical in a biological sense (and indeed during exactly this period, the Dark Ages, there was enough conversion to Judaism to upset the Church authorities enough to constantly issue declarations against the practice).
In this sense it might be better not to make it "Blood of" because with the Old Blood, Royal Blood, and Blood of Eagles you really are talking about lineal descent, and "Bloodless Xristos" plays nicely on this to suggest a fellowship which anyone, of any birth, can join, irrespective of ancestry. 6th-century Judaism is something in between -- definitely a familial relationship of descent, not an anti-familial monastic-based utopian community, but one with specific rituals available for joining up.
"Children of" or "House of" might get at that better. "Children of the Covenant" or "House of the Covenant" would be a possibility. Or "People of the Covenant".
When the real-world Dark Ages Jews are calling themselves bet yisrael/bais yisroel, the House of Israel (and perhaps occasionally am yisrael, the People of Israel), they're referring to a specific ancestor. Israel literally means the Stuggler-with-God, and it's the epithet Jacob wins, after his night-long wrestling match with what's usually (and in my somewhat idiosyncratic reading of the text, euphemistically) referred to as "an angel". A plain reading of Genesis 32:22-32 suggests it's actually God he's wrestling with: "I give you this name because you have fought with God and with men, and you have won.”
I mention this in case you want to make some pun on what exactly it is they are the house of, naming the ancestor by epithet, though nothing immediately jumps out at me there.
"Chosen" is also interesting, though more theological, and you want the emphasis to be genealogical, making it parallel to the various inheritance systems. (Interestingly, at least in Merovignian France, where Roman and Salic law were both in force, Jews were considered to be a special case of Roman law; where Jews went to court in disputes with non-Jews they had the right to adjudication under Roman law, while inter-Jewish disputes were handled under Rabbinical law).
The Temple is also an interesting option, but I feel like in a way it emphasizes precisely those things you don't want to emphasize -- dispersion and theology. The Temple largely figures in medieval Jewish thought as a symbol of loss and exile, of what we don't have any more. There's also a certain ambivalence in Rabbinical attitudes towards the Temple. On the one hand, it's central to the tradition that the Rabbis honor; on the other hand, the Rabbis grow out of the populist Pharisaic movement that grew up in opposition to Levitical/formal temple authority, and their writings quite often portray the Temple priests as dolts. One of the main Rabbinical images of the temple is that of Yochanan ben Zakkai slipping out of besieged Jerusalem, founding the first rabbinical academy at Yavneh, and teaching that what previously was attained by sacrifice in the temple could now be attained by deeds of lovingkindness and atonement. In other words, the Rabbis were most comfortable with the Temple when its rebuilding was positioned far off in the messianic future, as part of the world's final redemption.
Also, unlike the patriarchal traditions of Abraham and Jacob, the Temple doesn't really belong to all the Jews equally. If there was a "Blood of the Temple", it would be the kohanim and leviim, the Cohens and Levis, the hereditary priesthood and their cousin-servants, who the Rabbis seriously downgraded from a reigning ecclesiastical elite to a powerless group with a purely symbolic vestigial minor function.
"Blood of the Temple" would be analogous to "Old Blood" or "Royal Blood" in pointing to a specific subgroup of the people following that inheritance system as the hereditary elite justified by lineage; the problem is, within that subgroup, the hereditary elite had been already deposed for centuries, and the territory-based ideology privileging their role (sacrifices in the Temple) replaced with a portable ideology privileging the new, non-hereditary, technocratic/nomocratic elite (rabbinic exegesis and scholarship, legal rulings, prayer and ritual communal reading of the Law).
"The Law" might be another thing to work in there. "Children of the Law" would be another spin on "Children of the Covenant", pointing to the effective source of both power and unity within the dispersed Jewish community...