Your question is "is there a scenario when finishing with  would be considered out of place or context?".
As you noted, 宜しくお願い is similar to "Cheers" or "Regards", but the main difference is that neither of the latter are calls to action, whereas the former has more of a feeling of asking something.
Accordingly, among coworkers, it's fine to use when you're asking for something clearly scope of Things You're Allowed To Ask. I understand that may sound subjective, but that's part of the nature of the Japanese workplace: understanding your position.
On the other hand, if you're asking your boss to do something for you personally, it may be too direct as it implies you think that the other side will comply with your asking. In those more sensitive contexts, it may be better to say 〜して頂け[ますと/れば]幸い[です／に思います], literally translating as "if you did indeed do that, I'd be happy" without asking for it so directly.