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," meaning the same as "As I have got older, I have become (been) more confident?" 
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.