In the entry for "Melchizedek" in the Eerdmans Dictionary of the Bible (2000), we read the following under the heading of Psalm 110 (emphasis added):
Ps. 110 is generally classified as a royal psalm that lauds the position and attributes of the king. Textual problems, particularly in Ps. 110:3, 6-7, make it exceedingly difficult to reach positive conclusions. In Ps. 110:4 the king is called “a priest according to the order of Melchizedek,” but the precise comparison is unclear. It is far from certain that Heb. ʿal-diḇrāṯî (commonly rendered “in the order of”) refers to an official religious order per se, for elsewhere the phrase simply means “on account of.” To be sure, the oracle considers the king to be “like” Melchizedek, but again, the absence of data on Melchizedek as a historical figure precludes certainty about the precise meaning of the phrase.
This refutes the claim among some Evangelicals that, with the exception of Jesus only, no Davidic King held the priesthood “after the order/on account of” Melchizedek.
I have added the following quote from this standard scholarly work to my Response to The Narrows Church on the Melchizedek Priesthood