First of all, I'd like to note that few linguists or style-book writers consider split infinitives to be "poor grammar" today, and I don't know if there was ever a consensus among educated speakers against splitting infinitives, so I'm not sure it's accurate to say that they "became" poor grammar.
The origins of this supposed rule seem to be mysterious. Geoff Pullum says that
one source that may have been influential is an 1866 book called A
Plea for the Queen's English by Henry Alford (a churchman who was at
one time the Dean of Canterbury). Alford gives no evidence or argument
at all, but simply cites a prejudice of his own that is contradicted
by the testimony of another English speaker who wrote to him. He says
- A correspondent states as his own usage, and defends, the insertion of an adverb between the sign of the infinitive mood and the
verb. He gives as an instance, "to scientifically illustrate." But
surely this is a practice entirely unknown to English speakers and
writers. It seems to me, that we ever regard the to of the infinitive
as inseparable from its verb. And when we have already a choice
between two forms of expression, "scientifically to illustrate," and
"to illustrate scientifically," there seems no good reason for flying
in the face of common usage.
I have also heard the "based on Latin" explanation, but I've never seen any direct evidence for it. (I'm aware of the indirect evidence; that the infinitive in Latin is one word and so on.) If anyone has, please downvote this answer and provide the evidence; I'd be very interested in seeing it. Wikipedia also seems to agree with me (for what that's worth; I think the article is at least worth a read).
It seems to be true that Latin grammar played a role in some cases of 18th-century prescriptivism. The following paper gives a bit of an overview: "Prescriptivism and preposition stranding in eighteenth-century prose", by Nuria Yáñez-Bouza. It's about a different topic, preposition stranding, but it still may be of interest. Yáñez-Bouza says that in this time period "Forms not paralleled in Latin grammar were condemned as bad, incorrect, inaccurate, absurd, inelegant, or branded as solecisms." I don't see any direct references to Latin in the quotations she provides from 18th-century critics of stranded prepositions, but the paper cites various sources that might provide more information about the general topic of 18th-century prescriptivism and how it was influenced by Latin grammar. Since you said you are interested in learning about specific figures, you might want to look into James Harris (1751), who Yáñez-Bouza says was "one of the main representative figures in defence of universal grammar and analogy with Latin."
I'm not sure about attributing the "split infinitive" proscription to Latin influence though because early grammarians of English did use other criteria; they didn't just slavishly imitate Latin grammar in all of their descriptions.
According to "Early challengers of norms in the English grammatical tradition," by Henri Le Prieult, it is true that
"the first English grammars — from Bullokar’s Pamphlet for Grammar
(1586) to Lane’s Key to the Art of Letters (1700) — [...] display
evidence of the passage from the classical tradition of scholarship —
when Latin and Greek, often called the instituted languages, had long
been the sole tools and objects of learning — to the surprisingly
sudden designing and making of grammars of the European vernaculars.
However, according to Le Prieult, "throughout the early modern period, references to this tradition were remarkably bold and critical." Grammarians also made reference to established custom in English, and to principles of "nature and reason." Looking back at the Alford quote, you see that his objection is based on custom (saying the practice is "unknown" to English speakers and not "common usage"); it doesn't rely on Latin grammar in any obvious way.