I’m from the US and I’ve never heard this phrase before. I looked it up and its usage peaked in the 1940s. I would say exactly what you mean: “I don’t want to give you incorrect information,” or, if you want to sound more casual, “the wrong info.”

It’s pretty rare, and I wouldn’t recommend using it, but your examples aren’t good. I don’t want to give you a bum steer.

The phrase "to give someone a bum steer" is more common in the US than in the UK, but it's not used very frequently in everyday conversation in either place. It means to give someone incorrect or misleading advice or information. Your sentence is fine, but here's another way to say it: "Sorry, I need to check that and get back to you; I don't want to give you the wrong information." You can also say, "I got a bum steer," which means you received incorrect or misleading advice.

I've heard it. I use it. It may be dying out. It is definitely slang. It's usually a bad idea for English learners to _use_ slang. It doesn't make you "sound like a native" and it makes you _harder_ to understand, because we are not expecting to hear it. Here are some examples of use: "I'm sorry I'm late. I asked for directions, but I was given a bum steer." (I was given bad directions). "Where can I get some documents notarized?" "I think the public library does that, but you'd better call them and check--I don't want to give you a bum steer."

I have heard it used a lot. The answer you gave is correct, but "bum steer" is slang. Use it in less formal situations. In situations where you want your English to sound more refined, say "I wouldn't want to mislead you".