The priesthood of all believers also forms part of Catholic doctrine.
There is no ’rulebook’ in the Bible in the Gospel narratives, but that does not really matter: tradition and the magisterium supply in this instance.
In any event, you might wish to refer to Matthew 14:19
This kind of reliance on scriptural references is, you will forgive me for saying, quite a protestant approach though
I would suggest that if we're looking to traditions for what to do, then maybe we should look at the tradition in which the Last Supper was set, that being the Jewish tradition of Passover.
The meal of Passover is a family based meal and so there would be no (Jewish) Priest involved. Which would therefore make it quite a shift to require a Priest to be involved when taking the elements from it and using it in a Christian setting.
As to the Bible verse referenced, that was related to the feeding of the 5,000 and so although is an example of Jesus giving food to the 12 disciple, it is likely to have little to do with how the Last Supper should be delivered.
However, coming from a Protestant background I'd probably quite strongly agree with Paul at the end of Galatians 2.
Catholic Bible Study Online, Douay-Rheims Version. Complete text, index, search, Bible verse, bible story, scripture, book, prophesy, christian ...
In a more modern version (which is more in line with how we would speak now):
"Is it not clear to you that to go back to that old rule-keeping, peer-pleasing religion would be an abandonment of everything personal and free in my relationship with God? I refuse to do that, to repudiate God’s grace. If a living relationship with God could come by rule-keeping, then Christ died unnecessarily."
(Whilst many within churches may not like to quote such versions, for those without a church background it is far easier for them to read and understand; which is why I've included it here).