Hi Janet,
Not sure if it will get you any further forward but there is a death index entry for Thomas Bird, aged 57 in the September Quarter of 1903 [Maidstone District, Volume 2a, page 414]. Maidstone is the right district for a death in Staplehurst. If it turns out to be a Thomas who died in Staplehurst then the name of the informant may well be a relative and give you something more to go on.
There is a likely birth entry for Thomas Bird in the June Quarter of 1845 again in the Maidstone Registration District Volume 5 page 349.
Had a good look in the 1891 and 1901 census for both Matthew and Esther but, like you, could not find any potential candidates. Thought you might like to know in case you suspected you had missed something
.
I sympathise with your plight, I have a Great Grandfather who was a brickmaker and "travelled" a lot. He was born in Devon, married 1st in Somerset, married 2nd possibly in London and 3rd in Bethnal Green also having children to wives before and after marriage and using any combination of his three christian names! Still cannot find him in 1871 and I suspect he spent the night sleeping rough to avoid the enumerator - or am I becoming paranoid
Good luck with your search,
Paul