I was just doing a quick search when I came across this post. I am also searching the Bartle's family, also in Lincolnshire and have also found the name Francis in my line.
My great grandfather Thomas Bartle was born in 1877 in Beamish England and married Annie Mitford
His father George Francis Bartle was born in Deping Fenn Lincolnshire in about 1844. George F B married Anne Haley in Dewsbury Yorkshire and they had the following children: Thomas (as above) Joseph Elizabeth John G and Mary A
George FB's parents were John Bartle born abt 1814 and Mary Ann born abt 1819. I'm waiting on certificates from this link to enable further research. But if anyone thinks that this could be linked to them, I would greatly appreciate any extra information. Sounds like their were a few Bartle's in the Lincolnshire area???