Good explanation. Very plausible. Batlaao now seems old Urdu, perhaps from Delhi.
Both however are interchangeable when need arise..
~ Koi Batlaye ke Hum Batlayein Kia
And
Har Ek Baat Pe kehtay Ho Tum Ke Tu Kia Hai
Tumhi Bataao Yeh Andaaz-e-Guftugoo Kia Hai