It turns out "trimmer" was the term in that era for men who worked at a garbage incinerator: a conveyor belt would carry the collected trash toward the incinerator, and the "trimmers" would pull out things that shouldn't go into the flames -- things like mattress springs and other bits of metal that could be sold to scrap metal merchants, for example. Those merchants would pay the city for the chance to haul off this junk they could resell.
And in New York City, such an incinerator plant was located just off Tompkins Street! So we may assume "Thistle, the Tompkins Street trimmer" was one of those fellows, working at that location very near the Bowery.