Colour profiles don't make a difference when printing B&W as far as I'm aware. There are some obscure videos on YouTube regarding creating B&W profiles which get extremely in-depth.
The biggest factor will be paper choice, but using a manufacturers colour profile won't help with B&W printing.
You don't mention what printer you have, what paper you're using, or what the lighting conditions are in your printing area so it's very tricky to advise. My advice would be to get a box of A4 of whatever you want to use (or a test pack of different papers if you don't know), do a basic print with no calibration to see how it looks. When you compare to your monitor, do it in good light. A monitor is backlit, whereas a print is lit using reflected light. I use my conservatory for it.
Make notes on the print of areas that have issues. Usually it's very deep shadows and bright highlights. Adjust your monitor so that it looks like the print. Try again. If it works, and your monitor is capable, save it as a monitor profile for that paper. Rinse and repeat as neccessary.
I have found Lightroom's soft proofing to be next to useless because it uses the paper *colour* profile which isn't representative (in my experience) when it comes to B&W. I have found screen calibration systems also great for colour work, but not great for B&W. My options were "make the screen look like the print", or "go down the rabbit hole of building B&W profiles".
As for Printer choices, my Epson has a dedicated B&W mode which is excellent for reproducing what's on my screen. In my case, I can get away without adjusting the screen, instead relying on the fine tuning in the print module, and making a preset for that paper which I can reuse with confidence. I am extremely satisfied with my B&W prints.
Hope that's helpful, and I'm sure others will chime in with different replies.