Are you American? I'm guess no, because the term bit hasn't been used here in about 100 years.
So the slang "two-bit" (such as a "two-bit company") is not used in the US (thought I'd read some joke about Microsoft Windows years ago that had that term in it)?
Yes, that specific phrase is used occasionally to mean something cheap/worthless, but that's the only usage I've ever heard,
"i'm feeling a bit tired".
bit/bit/
Noun:
1. A small piece, part, or quantity of something.
2. A mouthpiece, typically made of metal, that is attached to a bridle and used to control a horse.
3. A unit of information expressed as either a 0 or 1 in binary notation.
I think "two-bit" company does refer to the monetary "bit", but that's the only usage in that sense I can think of too. It's just a bit further obviously refers to "a small piece".
FWIW I think the 'bit' in bitcoin is strictly to indicate something technical. Most people think of "bits" to be the things used by computers, as in definition #3.