Hello Mark,
You ask, "How can I investigate this in a way that will enable me to come out of it and remove this hindrance in practice?"
Has it ever occurred to you to use that anger, doubt, or lack of faith as an object of your contemplation in order to examine it so that you can see and understand its origin? And, perhaps, by doing so begin to let go of it?
Or is it buried so deep in your subconscious that you are unable to identify it in order to understand the cause and resolve its power to bother you? In other words, it is there, but you lack the self-honesty or the insight to properly accept that this or that is its cause. Without knowing the cause, you are fighting an invisible ghost! How much sense does that make?
Once identified, any hindrance can be dealt with and let go of. But not until. You would need to use a
vipassana (insight) meditation approach to do this. Similar to what sv monk is suggesting in his second paragraph.
Sit in contemplation and investigate whatever comes up. Examine it from every angle you can think of. Use that time to gain insight into the source of your disturbance. And once you find out what it is, become still, by letting it be... in order to still the mind.
There is a quotation from an old television show which may inspire you to dig deep within yourself.
Caine: I am troubled.
Master Kan: Why?
Caine: My parents are long dead. General Chung is tumbled from his arrogance and power. Yet within me anger boils as water in a heated pot.
Master Kan: Observe the day lilly. Each morning with the warmth of the sun, it opens in lovely blossom. Each night it closes.
Caine: Am I to do nothing, feel nothing, be still?
Master Kan: Still water is like glass. It is the perfect level. A carpenter could use it. The heart of a wise man is tranquil and still. Thus it is the mirror of heaven and earth, a glass of everything. Be like still water: you look into it and see yourself.
Just some food for thought; that is, if you care to contemplate it.
In peace,
Ian