Why doesn't James McGill join a different law firm?

I think the main reasons are:

1. Jimmy wants to impress his brother and
2. he wants to work with/for (especially) him (Chuck)
3. Kim also works for HHM (Jimmy has a special relationship to her)

The short answer here is because Jimmy wants to make Chuck proud.

As much as Chuck is a source of frustration for Jimmy; Jimmy spends his life in search of Chuck's approval. A short rundown of events:

• Chuck pulled Jimmy out of his criminal past. He gives him a job in the mail room of HHM.
• Jimmy gets a degree because he wants to prove to Chuck that he's making steps to improve his life. Chuck's achievements as a lawyer probably incentivized Jimmy to pursue the same career.
• Chuck outright dismisses Jimmy's achievement and practically ridicules him for it. Not only does he fail to acknowledge Jimmy's intentions, he actively points of that it's not a "real" degree.
• Jimmy pursues his career as an independent lawyer, as HHM isn't giving him a position.
• Jimmy starts taking care of Chuck when he gets sick.
• There are clear indications that Jimmy doesn't believe Chuck's illness exists, but he still chooses to appease Chuck. He does everything for him, effectively butlering for him at times.
• Chuck never really thanks Jimmy for the effort. He only asks more and more of Jimmy.
• Despite everything, Jimmy keeps going back to Chuck. Even after big fights, Jimmy still takes care of him.

Jimmy is devoted to Chuck, well beyond what is expected of him. And we don't quite see why that is the case. Jimmy isn't really benefiting from his relationship with Chuck, there's no quid pro quo. There are only two possible reasons for Jimmy to stand by Chuck:

• Gratitude/loyalty for giving Jimmy a second chance. This is quite unlikely, as:
  • Jimmy is shown to still think that cutting corners and conning people is okay, which means that Jimmy doesn't consider his past life as behind him.
  • Chuck is outspokenly negative about how Jimmy repeats past behavior, and Chuck sometimes jumps the gun about Jimmy before he even knows that Jimmy actually did something wrong.

Which leaves us with the only other option:

• Jimmy craves Chuck's approval.

This is actually a very realistic and common phenomenon. Someone who seeks validation from someone who specifically refuses to valide them, can cause the first person to pursue that approval against all odds.

A simplified example of this is negging. Effectively, negging someone is treating someone like you're Chuck and they're Jimmy. While you may initially think that this causes the "Jimmy" to dislike the "Chuck", it actually creates a challenge for the "Jimmy" to impress the "Chuck".

Note: Negging is defined as a pickup tactic, which of course isn't the case for Chuck and Jimmy. But the underlying psychology of craving approval is the same.

Note 2: I do believe that Chuck isn't intentionally manipulating Jimmy into seeking Chuck's approval. I think Chuck is simply being his usual arrogant self, and is unaware of how his behavior impacts Jimmy.
There is a scene in season 3 where Chuck seems to at least consider his actions from Jimmy's point of view, and he immediately becomes remorseful and on the brink of tears, literally stating "what have I done?". It seems fair to assume that Chuck is oblivious rather than malevolent.

On a more personal note: I have personal experience on being the Jimmy in this relationship with one of my parents. It took me decades to stop seeking the approval of someone who is unwilling to compliment anyone but themselves. I can personally attest that Jimmy seeking the approval of Chuck (to a fault) makes perfect sense. It is also not a rare occurrence, as I've had contact with people in similar positions than mine who all experienced the same behavior of idolizing their abuser.