apologies for the delay in replying but it was very busy over Christmas and there are not volunteers in all parishes.
I checked the 1901 census for a Robert Fulton and got 15 hits but one 93 year old living with his son and family stood out so may be your left behind relative. See http://www.census.nationalarchives.ie/search/ Select 1901 census as it defaults to 1911.
The Church of Ireland have records going back to the 1700s which they are slowly putting online and an email might find out if the Tyrone ones are available to go on line as not all records survived. See http://ireland.anglican.org/information/63
The Flax Growers list of 1796 gives a number of Fultons in Tyrone but no McLarens. See http://www.failteromhat.com/flax1796.php
Also this site may have some records if you emai http://nmni.com/uafp/Collections/Centre-for-Migration-Studies
Northern Ireland records are held here http://www.proni.gov.uk/
On the tithe applottments strangely there are no Fultons in Tyrone and I wonder if there is a slip here see http://titheapplotmentbooks.nationalarchives.ie/search/tab/home.jsp
There are 130 Fultons in Griffiths Valuatin circa 1850s see http://www.askaboutireland.ie/griffith-valuation/index.xml?action=nameSe...
As we appear to have no volunteer in the area a letter to a local newspaper may assist in finding family.
Please feel free to ask again if this does not assist.