Fine wirer Skilled pickpocket.
From the glossary of Graham Nown, Elementary, my dear Watson: Sherlock Holmes centenary; his life and times (1986) 1
You've established the meaning of finewirer - not a straightforward thing as it is an obscure word, verging on cant. The term seems to be used most often with an explanatory gloss or, in novels, with some nod to its abstruse nature.
“I was well on my way to being a fine wirer when I gave it all up. More's the pity! But I'm doing a better work now!”
“What in the world is a fine wirer?” Flora asked.
The young man held back his head and laughed. “You two don't know nothing!...”2
Clarissa Ross, China Shadow (1974), p97
'E said one boy as Mr Durban caught weren't five or six at all, 'e were more like ten, an' 'e were a right thief, 'alfway ter bein' a fine wirer. That's someone as can pick a lady's pocket an' she'll never even feel it.'
'I know what a fine wirer is. ...'3
Anne Perry, Execution Dock (2001)
Chaff told him to watch and learn because a cracking fine wirer was going to work. She smiled at that. Did Ash even know what a fine wirer was? She wound her way into the crowd with practiced ease.4
Nikki McCormack, The Girl and the Clockwork Cat (2014)
As to the word's origin, according to Stephen Van Dulken in Inventing the 19th Century: The Great Age of Victorian Inventions (2006), at p150:
There was a great deal of crime in Victorian London, and it has been said that the reason why so many thieves looked famished was that it was an overcrowded profession...
'Tailing' was the business of stealing handkerchiefs from back pockets. Pickpockets were dubbed 'buzzers' or 'dippers' while those who helped by obstructing pursuit were the 'stalls'. Those who specialised in stealing purses were 'finewirers' or 'maltoolers', a reference to their fondness for using (unpatented) devices to steal purses. The devices included slender blades to rip pockets, or tools which had a three-way gripping hold that could be inserted into pockets, and which cost 10 shillings (50p) a time from disreputable shopkeepers.
This makes intuitive sense, although the author might have a bias to seeing technological word origins, given the thrust of his book, and any sources cited are not apparent from what can be seen in Google Books.
This derivation doesn't necessarily contradict the suggestion you make - that there may be a link with wire in the snare an animal (with wire) sense - although it doesn't confirm any direct link either. Wire itself seems essential to both the trapping and the 'pocket-fishing' so finewirer and wirer (in the trapping sense) might well be completely independent of each other.
Having dealt with the wire, moving on to the fine: a finewirer seems to be a cut above the average cutpurse, the most sophisticated of their profession, and the fine probably refers to their high degree of skill rather than fine-gauge wire, fine meaning:
very precise or accurate...
delicate, subtle, or sensitive in quality, perception, or discrimination...
superior in kind, quality, or appearance 5
(Note emphasis has been added to quotes throughout.)
Your question about research methods in general
Apart from this word, how do you go about researching terms like this one?
is probably better suited to the EL&U Meta site where it will get a better answer than I could possibly give it. For what it's worth, I found all this information with a Google Books search for finewirer, and, as a starting point, this question, What good reference works on English are available?, is definitely worth a look.