I haven't developed many budgets for documentaries, but I have done a number of low budget and short feature budgets.
@Alcove Audio laid down the basics of budgeting when he said you can build a budget from the bottom up or shoehorn a budget into a known amount. In order to shoehorn, you really have to know a lot about producing films or docs, to know where and how you can shoehorn. So, if you really are set on a $10K budget, then as others have said, you need to partner with someone that knows how documentaries are produced to know how to realize your vision on a $10K budget.
So, you should consider spending your energy now to find the right partner.
If you really want to start on a budget, then start with the bottom-up approach. Figure out how many interviews you want conduct and how many days that will take. Then put a number to equipment rentals, crew rates, meals, transportation for the number of days you need. Figure out how much stock footage you think you'll need. That's probably a tough thing to estimate, but it sounds like you have some vision for this doc, so you might be able to say, you need stock footage to cover x, y and z.
No matter the project, developing a budget always demands that I do some research. How much will it cost to rent a cow? Make a list of the questions you have and things that need research. Then you need to call around, ask friends, come back to this forum and see if anyone can answer questions you have on specific items. For stock footage, call the tv stations and try to get a quote. Most things are negotiable, but you might be surprised if you are counting on licensing all your stock footage for $2,000 and the first person you talk to wants $10K, just for their piece.
Building a bottom-up budget is a great way to begin to address the logistics of what you are planning. More importantly, if your bottom-up budget comes to $57K, it might drive you to consider looking for other sources of financing.
My final thought. I think if you read about making docs or talk to other documentary filmmakers, you'll find that docs are uncovered in the process of editing. Features start with a script. Docs tend to start with a lot of footage and the script is written from assembling that footage. Most docs I know or have read about, started with an idea that morphed as the footage was assembled.