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.