Bruce, I have a possible candidate for your Elisha.
The locations of East Bethany and Batavia are tip-offs that this is almost certainly someone from the Connecticut Churchill family, which began with the immigrant Josiah in Wethersfield, Connecticut.General Worthy Lovell of this family settled in Batavia, New York, in the very early 1800's, probably in 1802.He was the first or among the first settlers.He was followed by a great number of Connecticut Churchill families who settled in the area, including Bethany and East Bethany.East Bethany is 6 or 7 miles southeast of Batavia, and Bethany is one or two miles to the southwest of East Bethany.
I thought this would be a fairly simple search, for though the standard Churchill genealogy, "Churchill Family in America," (CFA) published in 1903, has an index of family names but not of place names, my own book on the Connecticut Churchills has both. However,using both books and searching both by location and all the names you provided turned up nothing other than the one person I will discuss.This in a sense may be positive, for the fact that there are no other candidates increases the probability that the Elisha I found is the right one.
There was an Asahel Churchill born in Connecticut May 1, 1754 (no. 50 in CFA).His wife, or first wife, was Eunice Piney, who was the mother of his children.There has been a great deal of uncertainty about some of the details of this family, and they have been the subject of many postings on GenForum.The line of descent from the immigrant Josiah is Josiah, Joseph, Samuel, Benjamin, and Asahel.For your purposes the important thing about Asahel is that he had eleven children, the tenth of whom was Elisha.
A son of Asahel settled in the Black River Country of New York and was joined there sometime between 1805 and 1809 by Asahel and much of his family.This would have included Elisha, who was born August 16, 1798.
The Black River rises near Boonville, where other Churchills settled, and empties into Lake Ontario at Watertown.Though Asahel and his (2nd?) wife died in Jefferson County, where Watertown is located, much of his family moved on to the Batavia area.
Per GenForum posting 2216 it appears that Elisha's wife was named Louisa and that they, too, lived out their lives in Jefferson County.However, CFA has limited information on this family and shows only two children for which it has very abbreviated information.This very often suggests missing information, that is, the existence of other, unnamed children.This is made even more likely by the information from posting 2216 that both Elisha and Louisa had normal life spans, that is, that they were both alive throughout her child bearing years.
Moreover, though we do not know where the second daughter listed, Olive, was married, and though the first listed daughter Lovina's first marriage was to a man from Jefferson County, Lovina's second marriage was to an Allen Benedict of Fairport, New York.Fairport is just to the southeast of Rochester and thus relatively close to Batavia.Inndeed, sometime after 1865 a son of Elisha's brother Ara in Batavia moved to Fairport.
So, it's possible that your information that Eliza Jane Churchill was born in East Bethany reflects where she lived as an adult and that if you haven't been able to find evidence of Elisha or his daughters in East Bethany, you might find them in Jefferson County.