Hiring academics for very large industrial consulting projects - how to find an arrangement without full-time employees?
Professors at all levels of seniority and accomplishment are very busy.

They have courses to teach, institutional committee service, and their own research programs.  Neglecting any of these for any length of time can have a permanent impact on their career.  Professors who are less accomplished or have fewer duties in one of these areas will typically have more in the others.

Even a professor who is very motivated to work on your project may only have a few hours per week to devote to it.  One day per week would probably be a maximum.  Teaching schedules are often inflexible, so it may be particularly hard for them to visit your site, especially if it is far away from their home.

Senior people (tenured full professors) may be more likely to be able to make time for your project, but they also have lots of other interests.

One thought: you might consider approaching an expert who has recently retired.  They should have the time available, and may be interested in your project as a way to continue their work in the field and earn some extra income.  But they also may not be interested in working a lot of hours per week, and they may demand a fairly high hourly rate (as they're probably not in desperate need of money).

Keep in mind, also, that research by its nature is very hard to predict.  It may be difficult or impossible for your consultant to accurately estimate how much time they will need to solve your problem, unless it is so straightforward that they can solve it almost at once.  

As far as fresh PhDs, note that a new PhD has just spent several years working long hours for low wages.  It may be hard to entice them with an offer of unstable, part-time work, unless you can offer some other perks like stock options or an unusually enjoyable work environment; startups are a dime a dozen.  Of course, a lot will depend on the overall strength of the job market in your industry.

Your third option ("research arrangement") doesn't strike me as particularly feasible.  A major reason for being in academia is the ability to choose the direction of one's own work.  I don't think you'll find many professors amenable to being told how to direct their personal research program.  They not only have to publish papers, they have to publish good papers which are novel, interesting to the research community, and spur further work in the area.  If your project doesn't lend itself to that, in the professor's view, then the right to write papers about it is useless to their career and won't be seen as a benefit.  Moreover, if they're going to work on your project as part of their own research, which is on their institution's time, then you can expect the institution to get involved in any IP issues, and they often have lots of lawyers.  Not to mention outside funding agencies.