I don't quite understand how your company can prevent you from providing a "personal" reference for someone.
As for not getting or using a "professional reference," it's done all the time. I did not use my last employer as a reference for my current position...although the people I work for now, know I worked at the last place of employment.
Reason being: The boss and I had somewhat of a personality conflict at the time, and I wasn't sure I wanted to give her a reason to say anything unflattering...and certain personal comments had come back to me, from the "grapevine." (Though my work was not in any way, anything she could badmouth. It was actually the first time I'd been "fired" for doing a "good job." She just couldn't afford to keep me on.)
It's been three years since I worked for this woman...and it's taken her all that time to finally look me in the eye and speak to me, without slinking past me like I'm going to deck her! We're actually on good terms now. (Not that we weren't before, but she was needlessly intimidated then, and has found out since, what a silly nut I really am.)
We get along reasonably well now, particularly in social situations. But I still wouldn't use her as a formal "reference." Yet, I have interaction on a regular basis with that office, in my current position, and have a very pleasant relationship now with she AND the staff there.