It is true that Jesus did not have the Aaronic priesthood, but didn't he posses the Melchizedek priesthood and thereby could offer up himself as a sacrifice?
The claim that Jesus held a priesthood "according to the order of Melchizedek" is made in Hebrews 6:20. This claim is irrelevant to any discussion of sacrifice made under the Mosaic Law. It is that Law that Jesus is said to have fulfilled and nullified by offering up himself (Hebrews 7:27). The so-called "better covenant" (Hebrews 7:22) allegedly instituted by Jesus and its accompanying everlasting Melchizedek priesthood have to do with Christian beliefs concerning Jesus after his death. A non-Aaronic order of priesthood has no relevancy to the requirements of the Mosaic Law.
The New Testament claims that Jesus fulfilled and nullified the law by offering up himself as a sacrifice. According to the description it provides, this came about in a manner that runs counter to the Mosaic Law. Yet, the claim is made that everything Jesus did was in accordance with the Mosaic Law. This is a Christian conundrum. To the rest of us, it is obvious that this Christian claim has no basis in fact