First of all I checked the General Alphabetical Index to the Townlands and Towns … of Ireland (1861) to check the name of the parish in which Ballynashee is located. It was Rashee. Ballyclare is in the Grange of Doagh. Nearby are the parishes of Ballynure and Ballycor.
Reviewing the sources that are available for these parishes, I would recommend that you check or have checked on your behalf the following records:
Estate papers
Until the late nineteenth century, the vast majority of farmers in Ireland were tenants on an estate, paying rent to a landlord for their farm. The principal landed estate in the area you are interested in was the Marquess of Donegall’s estate. A large collection of records relating to the management of this estate can be inspected in the Public Record Office of Northern Ireland (www.proni.gov.uk). These records include, maps, leases and rentals.
Church records
Unfortunately there are no church records for this area prior to the nineteenth century. You can find out which records do survive from the early 1800s via the website of PRONI.
Census substitutes