Replying to and old topic but here goes:

I learned a similar story in Pediatric Residency. As I learned it, toddler derives from TOT (a small measure of rum). When one was a TOTTLER, one drank too many tots and stumbled. An infant was a small measure of an adult and also stumbled like a tottler. Hence the terms Tot and Toddler.

Google searches through the decades have failed to find supporting evidence, but it is still a fun story to tell new parents (with a historic-accuracy disclaimer)