1)As I have got older, I have become (been) more confident 2)As I got older, I became more confidentLet me present my wild guess to help you understand my question (I made the second example...

Your verbs must agree, and that sense of agreement has everything to do with tense or time.  So, while it may seem nit-picky, you will need to correct both statements so that the verbs agree as regards their tense.

Let's look at your final question:

But I am wondering if I can say, "As I got older, I have become (been) more confident,"[2] meaning the same as "As I have got older, I have become (been) more confident?" [1]

First, both statements don't agree in tense, so they must be corrected before we can address your question.  You have written:

As I have got older, I have become (been) more confident.

You must make sure that the tense of your verbs agree.  The statement should read:

As I have gotten older, I have become more confident.

The same is true of your statement (2).  To agree with its own tense, it should read:

As I got older, I became more confident.

The short answer as to whether these two statements (1 and 2) are interchangeable relates back to the tense of the verbs.  So I would say no, that because you are creating different tense structures, the statements are different in meaning since they describe time differently.  Your statement 1 describes action that happened progressively, while statement 2 is a observation of the simple past.

For more on verb tenses and how they distinguish time, please follow the link below.