Buying domain variations to increase local seo

So here's a question I pose to those who are way smarter than myself..

ABC Business Company, (ABCwebsite.com)
Local business with multiple locations: e.g.
ABCwebsite.com/locations/city1
ABCwebsite.com/locations/city2
And so on.

ABC Business Company is also known as ABCBC. it is so commonly referred to as ABCBC, that the company legally registered the name.

Should they buy the domain ABCBC.com and build that out - maintaining both websites? I feel like that could be as hurtful as it could be helpful.

In my thinking, they could increase visibility and conversions, and essentially create a ranking competitor - which would be themselves.

On the other hand, could Google look at that as violation and smack both sites down and issue a penalty?

Please, anyone with experience chime in, I can't be the first person to think about this.

Thanks in advance!

Have the name on Title and Meta Tags so it all get optimized for a single website - Problem Solved