After a quick Internet search, because I assumed that Rudy was capable of that himself, since he composed a decent e-mail, I turned to local historian & newspaper man, Tim Tefft. (Yes, that is his ancestor is mentioned above.)
Tim's answer was... interesting. Basically it comes down to legend & conjecture that will require further research, but here are the hypotheses...
Rudy's: that the river is named after the bats found in a cave near Mt. Aeolus near Dorset, VT. This cave is near the headwaters of the Battenkill. Rudy admits that to be plausible it would take a combination of Dutch & English etymology. (Tim points out that very few rivers are named for points near the headwaters, but for where they flow into another body of water.)
Tim sent a long e-mail with a couple of theories as well. One states that the river was named after Bartholomew Van Hogeboom, an early settler of the area. The nickname "Bat", plus the plural "en" & "kill" for river, give rise to the Battenkill name. However, Tim states
Now, it should be recognized that the Dutch word for the flying mammal is not "bat," but "vleermuis," derived from the Germanic "fledermaus" (or literally, fluttering mouse). "Bat," on the other hand is from the Middle English word "bakke," which has a Scandinavian origin. Dutch Americans, in the years after the English took control of New York, naturally adopted English words into their vocabulary, and vice versa. In this regard, they might well have adopted the English word "bat." Dutch formed their plurals in most cases by adding "en" to the end of the singular noun... Thus, the plural for "bat" would be "batten." And the river that formed at the Dorset bat cave that flowed all the way back to their homes in the Upper Hudson Valley would, of course, be the Batten Kill.
Next Tim mentions our local Bottskill Baptist Church, & that is name seems to derive from "Bart's Kill."The local hamlet of Battenville (where Susan B. Anthony's father moved the family in 1826) is named after the river, right? Or is it named after the "battens" used in looms?I can find no proof that Bart or Bat Van Hogeboom ever lived at the mouth of the Battenkill, but genealogies list a number of Bartholomew/Bartholomus Van Hogebooms who were born in towns along the Hudson from the early to mid 1700's, one of them was baptised at Albany in 1702. There is a tradition of an early 1700's community along the Hudson near the mouth of the Battenkill. If it existed, perhaps Bart Hogeboom was among the settlers.
What about the native name for the river "Dionondehowe". Apparently the earliest this name shows in writing is 1709. & is loosely translated as "she opens the door for them". Check out some amazing images of the Dionondehowa Falls here.
One thing is for sure. The Battenkill River is redundant & incorrect. Simply the Battenkill will do.