Is your God contingent upon your brain or does he have some sort of external existence from your brain?
God is not a physical being. My brain is only physical. And God created my brain. So, yes He is external to it, but also in and through it so it functions as well as it does.
Can I prove this to you? No. But God can prove Himself to you. I have needed how God has corrected me so I have become able to experience Him, because He is so superior to us, in love's character. As much as I am still selfish and about myself, I fail to do well with God in His peace and how He would have me loving any and all people and being able to relate intimately with people,
"without complaining and disputing" (in Philippians 2:14-16).
I did not get myself to become this way

This is better than my own selfish nature could get me to imagine. And your character, too, is your dictator of what you are able to think and do; only God is able to deeply change us, deeper than our brains and our own ability to think.
But if your own ability to think is your god and imaginary friend . . . you are limited by yourself. This is why Jesus says,
"If anyone desires to come after Me, let him deny himself, and take up his cross daily, and follow Me." (in Luke 9:23)
Jesus is not conceited. Even though Jesus is so superior as God's own Son, Jesus came to us and suffered like He did and died for our sins on the cross. And He is able to see how we really are, better than any friend can. And, of course, an imaginary friend can be so limited . . . to how the person is, who created that imaginary friend who can't give the person real correction of character.
So, God is so better
Does God exist in any way which can be verified to be more than just mental processes?
Well, He verifies Himself. It is better than how you need to have a person speak for oneself, rather than you trying to represent that person. But if you refuse to hear someone out, or if you are deaf . . . you can not listen to what someone has to say, plus you will be stuck with whatever conclusions you draw without actually relating with the person.
But in case you are deaf and unfeeling to God, because of your own nature making you so, God is able to get through to any person.
People can believe all sorts of things.
yes
And you and I have not been somehow superior to anyone else, as humans. We have been able to fool our own selves and not even know this. This can include how we fool ourselves into seeking what is not real proof of God. God proves Himself better than how we can humanly dictate that He proves Himself
It is like how academic people would not believe there was such a thing as a platypus. Even when they saw one, still they insisted that someone had sown parts of animals together to make the platypus. They needed to actually handle and dissect it, in order to know.
They had a point, not believing what other humans told them, though

There are humans who indeed will try to fool us. Plus, we can fool our own selves and not even know this . . . while we are so worried about someone else cheating us. Our own worry, for one example, can deeply degrade us and abuse us and lie to us, so much more than any other humans have ever done; yet, ones keep trusting and even obeying their worry!!!!
This is why we need God.
And God is the Creator of the platypus and of us. So, there is a lot more to Him than humans can deal with . . . or imagine. So, asking me to prove Him is asking for what is inferior

And so of course you can pick apart what I offer.
Before we can say that, we must determine that God exists and is not simply a mental construct.
Well, in case He does exist and you are managing to keep yourself from experiencing this, what am I supposed to do, more than offer all I have already experienced and found in the Bible? Why would you depend solely on humans for what we can do?
"I take it you can provide evidence this is true, that anyone else could examine."
If someone else still does not have the spiritual ability to experience how God is, in His quality of love, how can I prove this to that person . . . if the person is so incapable?
There are people who always have been light blind; how can I prove to the person that light exists???? A person told me she is blind and does not dream; how can I prove to her that there are dreams????
God is so better than all which I was imagining, and I needed major correction, just to get started with Jesus. And now I need more and more, so I can find out how to please and share with God and love any and all people. And the Bible helps me see how much more and better God is and desires to share with us.
HOW does God respond to you? Be specific.
Well, He doesn't really respond, but He first gets through to me and corrects me, then guides me in His peace. I need how God makes me capable of submitting to Him in His peace > Colossians 3:15.
At a practical level . . . He confronts how I can be paranoid and lusting and self-righteously looking down on others and picking and choosing who is good enough for me to love. And He exposes me to how He is gentle and kind and sweet and pleasant and caring and compassionate and generously forgiving and creative in how He has me find out how to love people. And He can share this with me, spiritually.