For the budget? Go digital, but don't worry about those monochrome sensors. Sure, they're strictly better in terms of per-pixel awesomeness, but even entry-level DSLRs and mirrorless cameras have high enough resolution to make up for it — especially if your comparison point is 35mm film. And the color information let's you easily make color filter choices in RAW conversion rather than before each shot with a physical filter — and with much more flexibility and nuance than a whole bag of filters.

You could go analog and set up a darkroom, but overall, the cost per shot is just plain going to be orders of magnitude higher. That doesn't mean it doesn't have its own appeal, and I expect that film (and pre-filter processes) will outlast digital as we know it (just as vinyl records outlasted CDs). But for the budet? Digital wins.

If you have a lab develop the film, you're looking at something like 25¢ per frame. And that's just developing — add that again for high-quality scans, and probably again for high-quality prints. The cost per frame for digital is effectively nothing per frame.

You can do it much more cheaply if you have your own darkroom. For black and white, that can be fairly cheap, assuming you have an adequately ventilated space that you can make light-tight. Still, you'll easily spend the difference between a digital body and a nice analog camera. Your cost per frame will be dramatically reduced after that, but still something like pennies as compared to that effective 0. And that's not even considering any option to scan your results — pure analog.

If you have the time, space, and interest, I wouldn't discourage you from going analog at all. But only then! If you're interested in getting great black and white results, you can do it either way. And if you make money your deciding factor, digital wins, and, really, the cheapest RAW-capable camera you can get your hands on.