For a department focused more on industrial interactions, will going to work in industry for a while before applying for a faculty position in that department be recommended/helpful?
I would think that you should consider a somewhat different question. What is the best path for me, now, to obtain a faculty position later? You probably have several options and each of them takes some time to implement. Which is the best path if the goal is to become a productive faculty member?
The two main pathways would, perhaps, be more education and some time in industry. Either can probably be good, but I would suggest that further/deeper education is probably going to be more valued than industry experience, with one exception.
If you were to find a unique, cutting-edge, industry position that puts you in contact with top-level research and development in an area that is seen as vital at the time, then that would be a nice thing to brag about on your CV. But working as a drudge in some mid-level outfit won't really help you advance in academia. These top-level positions are pretty rare, and may not be open to you with your current background.
Note that the top-level industry experience I'm mentioning is the sort of thing that you can bring back to the university to build a research program an/or give important advantages to students.
I think it is easier to find a way to stand out in the educational realm if that is where you want to be eventually, than it is to go and then transition back with the same level of excellence. But that is more opinion than fact, of course.
If you describe the field a little more, people can advise better. Perhaps there are still generalizations (helpful to others), but also you will get some tailored answers.
But I will say in general that even in rather applied subjects (chemistry, geology, engineering, etc.) that it is the norm not to have much industry experience. Even in topics like business. Doesn't mean it will hurt you per se. Or that there is nothing leverage-able. But the Academy seems to value the Academy (not saying they are right or wrong to do so).