What is the prototype of “Place blame where it is deserved / Blame where it’s due / Blame only where blame is due”?

There is a popular idiom to: "Give credit where credit is due". It would seem that some of those instances you quoted are derived directly from that phrase, substituting 'blame' instead of 'credit'. "Place blame where it is deserved", on the other hand, sounds like a more literal application than a variant on the idiom.

Use of 'lay'

"Sir, "I do exceedingly thank you for the remembrance; but am sightless of the "wrong, that was done to my lord H. for he had been gone two hours before I "knew of his being here. I will satisfy him of my innocency, and he shall fee I will lay the blame where it is due. I pray you fend mr. Eure to me on Friday morning, and I will both give him a letter to his brother, and satisfaction touching his nephew. I rest "March 30. ** Your true friend, "E S S E X." -Memoirs of the reign of Queen Elizabeth, from the year 1581 till her death. In which the secret intrigues of her court, and the conduct of her favourite, Robert earl of Essex, both at home and abroad, are particularly illustrated. (Published 1754)

&

Waste not the time in fruitless complaints of this misfortune, or that accident; this minister, or that commander ; but lay the blame where it is due, upon national wickedness, which has called down national calamities. -Sixteen sermons on various subjects and occasions ... . Horne, George, 1730-1792 (published London, 1795)

Use of 'lodge'

Let our representatives in the legislature bring the question of their country to a vote, that they may give some proof of their own Zeal; and if we are to be disgraced, that they may lodge the blame where it is due, even upon us, if, after a fair inquiry, we shall appear to deserve it: that if we owe our mortification to a want of confidence -The Scots magazine (1762)