Transactions of the Cumberland & Westmorland Antiquarian & Archaeological Society is a long-standing local history periodical. Volume 9 (new series), 1909 has been scanned by Google (see here) but is not available for full view. An extract of potential interest is visible in 'snippet view' only:

"....value of the rectory (without the parsonage house) in 1719.
Before the death of Mr. Barton the Quakers had begun to refuse to pay tithes, &c., and Mr John Barton (brother and executor of the Rev. William Barton) had to take...."


Further searches reveal the following extract from Records relating to the Barony of Kendale, Volume 3 (1926) ed by JF Curwen, pp200-207, (online here), which must surely describe the same event, but more completely:
"1719/20 6 February. Whereas upon complaint in writing made unto us John Archer and Anthony Askew, justices of the Peace, by John Barton, gent, executor in the Will and Testament of William Barton, late rector of the parish of Windermere, concerning arrears in tithes from the Quakers, etc. On 5 December, 1724, similar complaints made by William Crosby, rector. And on 24 April, 1731, and the 11 December, 1741, similar complaints made by Girlington Butler Barton, rector of the parish of Windermere. Browne MSS., vol. xiv, pp. 83, 84, 85, 86 and 91; vol. iii, n. 55."

So it does not appear to concern the Ivegill Bartons, unless John Barton (d.1720)'s brother William went to Windermere and these events took place shortly before John's death (a slim chance).

Descendants

In February 2012, DBHB heard from Eric Butcher, a descendant of Girlington Butler Barton. Apparently Girlington was born in Lancaster, so is more likely a member of the venerable Lancastrian branch than a relative of the Ivegill Bartons (who can not be tied to that family. Yet).

In January 2014, DBHB heard from Jane Harrison, also descended from Girlington Butler Barton.