Dear Aaron (if I may),
There is, to my knowledge, little to no writing on this, so the following are the most basic of leads (and I'm afraid may not be that helpful) but here goes:
You may already know that the earliest textual history of the Rāmacaritamānas was published by Mataprasad Gupta in his Tulasīdās : eka samālocanātmak adhyayan (1942); he of course doesn't mention the Lavakuśakāṇḍ.
Imre Banga at Oxford has recently been revisiting the textual history of the Rāmacaritamānas; he may have some leads.
Biographical and bibliographical information for the scholar and translator Pandit Ramanarayan Pandey may provide some leads; he edited one of the editions containing the Lavakuśakāṇḍ and was incidentally a prolific translator into Avadhi/Hindi from Bengali.
On that eastern connection, the Lava-Kuśa narrative enjoys wider circulation in Bengal and Orissa than it does in Avadh; though I'm only speculating on a possible connection here, you may want to look at William Smith's “Variants of the Lavakuśopākhyāna” In Categorization and Interpretation.
All best for the search,
Tyler Williams
UChicago