And there's not much I can add that you don't already know I'm afraid. I have saved this line for a rainy day as it is not one where I had any relatives to talk to about it. I can give you the following snippits whether they are any help is another matter...
If there is one line and Lavinia is in it, then I'm pretty certain that Lovehannah is her, her grandson lives with them for both the 1861 and 1851 census (I don't know who the child belongs to), he called his daughter Lavinia, the Cowleys also use a corruption of the name with daughters called Vinia/Vianer and Vina. On the 1851 census Lavinia is down as Vinah and the troubled ancestry transcriber has mis-written this as Sarah.
I was simply going to do what you have done and trawl and list any Cowley/Burnside mention in Cockerton, Brafferton and Aycliffe. I was also going to trawl Haughton Le Skerne as being Lavinia's parish of marriage. Lavinia and George Cowley are on every census in Walworth which could be a possible avenue of research also if her family followed her. The Cowleys stick around the Walworth and Piercebridge area until they migrate to Billingham, it looks like the original George Cowley and Lavinia die in Walworth though and don't make the trip but i have no death dates for them as yet or evidence of them beyond 1861. The remaining Cowleys spend time in Carlbury, Coniscliffe/Piercebridge and then move mostly on mass by Nov 1862 to Billingham and stay there, have a thousand children and are probably now related to every one there judging by the number of marriage certificates I have.
I also think that the grave I mentioned could well be the Burnsides further down the line, I have been through every Billingham record and there is no Burnisde presence there except for these two burials, that plus the reoccurrence of the names Joseph and Robert is suggestive of maintained family links across the distance.
The only other relatively useless thing I have is two logged possiblities for Burnsides found on the census accidently when I was looking for something else, they were: Elizabeth Burnside on the 1841 census living at Bondgate in Darlington with Ambrose aged 18 an ag lab , and Matilda aged 16 a farm girl. The other was logged as Mary Burnside aged 51 in 1851 and was the toll bar keeper ay Gainford, she has a visitor from Piercebridge stauing weith her and a grandchild from Cockerton. In 1841 she is an ag lab living with daughter Jane in Denton.
This is all I have at present I'm afraid.