Don't rule out the Scamp connection - by following that family both backwards and forwards you may find the people you are looking for nearby on a Census or baptising children at the same church.
I've been looking at the information you have given and looking at all the data that might tell us more about the family but haven't come up with what I was hoping:
You say that "Elizabeth is on the 1939 census with Ernest in Faversham." - are these the Ernest Lee dob 16 Mar 88 and Elizabeth Lee dob 14 Dec 83 in a tent at Ulcombe, Maidstone in Hollingbourne RD in 1939?
There is a second tent with a Barrett family - look at these as there may be some relationship between a member of the Barretts and Elizabeth or Ernest.
"Annie married Abram Elijah Lee-Frost in 1926 in Sittingbourne Kent" so they should both be on the 1939 register - ah I've just found them as Frost:
10 Newbridge Avenue , Sittingbourne and Milton U.D., Kent, England
Abram 26 Feb 1904 - not 1903!
Annie 12 June 1909
William L 19 Sep 1927
Betty L 22 Dec 1928
Mary AL 28 Jan 1931 becomes Ditchburn
Abram is a Labourer at the cement works
"Abram was born 1903 in Kent" - where do you have this information from as his yob appears to be 1904:
1911 19 Crown Quay Sittingbourne, Sittingbourne, Kent, England
William Frost 1886 born Kent a labourer
Mary Ann his wife 1887
Abraham 1904 born Sittingbourne
Lennie (female) 1906
Levender (female) 1909
Abraham EL Frost dies 3rd Q 1965 Sittingbourne reg dist age 61 yob 1904
"On the marriage certificate it states Elizabeth's father is James Lee and was an Agricultural Laborer." - information about parentage on marriage certificates is not always correct - if one of the parties didn't know who their father was they would sometimes invent a father.
"Annie Lee was born 12 Jun 1909" - We need to find Annie and her mother and/or father in 1911 or on some other document.
If Ernest Lee married Elizabeth Lee in Herne Bay in 1910 where are they when the 1911 census was taken? Their marriage is June Q 1910 Blean Reg Dist. but I can't find them in 1911 so far.
FreeBMD shows Annie Lee married Abram Frost Sept Q 1926 Milton Reg Dist and there's a birth of an Abram A Lee Frost 3rd Q 1936 Milton Reg Dist, mothers name Lee.
Have you tried looking for any of these people in newspapers?
