Professionally, I mainly work with domains and brand names. And I'd advise 2 things:

(1) Whichever domain you're using to rank #1 organically for the desired search vertical, leave it alone. Building a practice on a geographically specific domain that also defines your profession is something I often recommend, boring though it may seem; and since the site is performing well (i.e. ain't broke) don't fix it.

(2) Rebranding may still be a good option for you; and since your instincts incline you to consider it, then you're probably right for the reasons you yourself outlined. When you choose a new name, then I'd recommend creating a separate site for it online. Eve if it's just a simple, inexpensive page, you can expand it later. That way you'll be transitioning with zero risk of upsetting your existing site's #1 position; and you'd be adding an additional asset to your business.

Now let's talk about that new name!