I engaged a company to do it for me on a project basis. Each “batch” of people I wanted to target generated a spreadsheet with their LinkedIn, email, phone (all of this pulled from reaching out as “me” on LinkedIn) and a meeting time. Using my personal LinkedIn generated an astonishing 87% response rate, which I don’t think can be replicated .. assume 5%-10% more likely. So, if you want to have 10 conversations next week, send 100-200 invitations. Short answer: I’d outsource it, with very narrow constraints.
Linkedin Sales Navigator should be able to help.
It's just a bunch of work somebody needs to do it. There are two approaches. One, where you go through your connections from LI ( You can probably figure out better than anybody who is likely to be more helpful than others in your connection list for a particular account ). Or mass outreach approach through outsourced vendor as suggested above.
If I remember right LI navigator helps figure out who is connected to who beyond your immediate network. This helps cases where you know people but just not connected to them on linkedin.