In this exchange, Tom and Huck are speaking to each other using youth slang of the 1860s, so it is understandable that when reading it 150 years later, some things don't translate. That said, I believe Huck has combined two idioms that are still in some use today.
For the first part, crowd through, I find in Idioms by Free Dictionary:
To push or force one's way through a certain thing or area.
This is most commonly used in a sense of concrete, physical obstacles, i.e. to crowd through the long lines at the shopping center during the holidays. Here, Huck seems to be thinking of more abstract obstacles, i.e. suppressing his urges to cuss and smoke whenever and wherever.
For the second part, or bust, I turn to Merriam-Webster:
informal —used to say that one will do everything possible to get somewhere
This idiom was familiar in America just a few decades ago, in the form of vacationers painting their destination on the back window of the car, for example, "Grand Canyon or Bust." This let all other cars passing by know where the occupants were going, and that they were not going to let anything stop them.
Putting it together, I might paraphrase Huck's portion of the dialog like so:
Ok, Tom, if you keep your promise to prevent the widow from harassing me about every little thing I do wrong, then I will in turn promise to not do quite as many things wrong (as least not in public). Further, I will make every possible effort, forcing myself to as necessary, to stay out of trouble. Now, about this gang idea of yours...