Looking closely at the the letters written by Theodore we can infer:

• He has a father named Walter Barrell, alive in 1799
• He has sisters Abby, Polly and Charlotte
• He has an uncle Theodore
• His mother is deceased in 1799
• He has an uncle Joe, who has a daughter Hannah
• He has an uncle Colborne
• He was educated at Mr. Andrew Rae’s Academy, Islington, London

It is likely that Theodore is the Theodore Barrell listed at https://barrell.one-name.net/getperson.php?personID=I40&tree=T014

A description of archives at Columbia University “Barrell family papers, 1751-1929 bulk 1791-1889”  http://www.columbia.edu/cu/lweb/archival/collections/ldpd_4078972/ says:

The Barrell Family were noted merchants and businessmen in London and in Colonial Demerara, now part of Guyana. Theodore Barrell, the son of Walter Newberry Barrell, a London-based businessman, was both in Boston in 1771. He spent most of his career working as a merchant in the Americas, including in Barbados where he met his wife Elizabeth Beckles Barrell (born November 18, 1783) and in Demerara. Theodore Barrell died in Saugerties, New York in 1846.

Theodore Barrell has a brief biography in Bram Hoonhout’s “Borderless Empire : Dutch Guiana in the Atlantic World, 1750-1800”.

The New-York Historical Society hold a “Letter book, 1798-1803” by Theodore Barrell – https://bobcat.library.nyu.edu/primo-explore/fulldisplay?vid=NYHS&docid=nyu_aleph001507649&context=L  described as

Summary: Letter book, 1798-1803, with correspondence to business associates in London, Barbados, etc., as well as letters to relatives and friends on personal affairs, and life in Guiana. Includes many letters to William and Samuel Jones, London; Walter Barrell, London; William Gill and Samuel Went, Barbados.
Language: English
Publication Date: 1798
Description: 1 v. (182 p.)

Looks like I need to make a trip to the Columbia University archives!