I have used LSP in counselling settingsfor a few years now with especially difficult participants (including post-trauma patients and a group of unemployed), and am familiar with the issue you present.
LSP is a great tool in this respect, rest assured.
The difficulty I see lies elsewhere, namely in hierarchy. Things will work out OK if you all are on the same hierarchical level, not if someone is above.
With hierarchy involved, reactions may focus more on the inevitable personal judgment the more powerful player will have after the session rather than on getting to a common solution.
LSP in a counselling setting can go very deep in a very short time and while your colleagues may not be fearful of not appearing professional enough when revealing their deeper motives in front of a stranger, they may in front of their boss-turned-relational-facilitator, which hopefully does not seem to be the case.
I usually have a longish heating-up part, where beyond the basic warmup people get to build and explain their own “avatar”, then their workplace, then their workplace with colleagues and themselves, and at this point you can introduce the specific problem they are experiencing.
Phenomenological feedback will go a long way, of course.