1 I don't like him because he stole my bike.
Correct. "because" refers to "cause and effect". It's the reason you don't like him.
2 I don't like him in that he stole my bike.
A search for "in that" brought up this unusual example: "But f1 is slightly different in that we did not use the." Still... it's sufficient to explain. "in that" is clarifying the first part of the sentence. "In what way is f1 slightly different?"
Some definitions of "in that" say that it means "because", however that overly simplifies it. Merriam-Webster writes "used to introduce a statement that explains or gives more specific information about what one has just said"
Therefore, it's not purely "cause and effect" like "because". Rather, it is explaining "in what way". This doesn't really apply to the bike example.
3 I don't like him inasmuch as he stole my bike.
Generally incorrect. There are a few reasons.
inasmuch is an unusual word. It's advisable to choose more common words such as "because", unless there's a strong reason otherwise.
The word "inasmuch" has various definitions. One of the definitions, which isn't the most common, is "because". If it clearly meant "because" then perhaps it would be acceptable, however the other definitions usually come to mind when hearing the word:
to the extent that; insofar as.
"these provisions apply only inasmuch as trade between Member States is affected"
considering that; since (used to specify the respect in which a statement is true).
"it was not really a still life inasmuch as all the objects were in motion"
- Therefore, the meaning of the bike sentence becomes something such as: "I do like him. He has many good traits. However, considering that he stole my bike... I can not like him any more." It's a comical or offbeat way of saying it.