I was born for this question.
No, really, you try going by a french name in English-speaking territory. Since I've been 5, the majority of people I encounter cannot pronounce my name. So I have considerable experience dealing with this. But wait, there's more! My birth certificate has both of my parent's last names. That's right, Jean-Bernard Pellerin is a dirty alias, my legal name is Jean-Bernard Landry-Pellerin. And it's not even that simple! As far as my SIN and the Canadian health care system are concerned, my name is simply Jean-Bernard Pellerin.
You try sorting that out at tax time...
As early as elementary school, I went by JB. Once I started University in a larger town far away, I tried going by Jean-Bernard, since like you I was concerned with the need for professional appearances. In the end, after the first few mispronunciations I just reverted to the abbreviated moniker and have not heard a single peep about it. I even met someone who goes by TJ and his name doesn't have a single T in it.
The actual solution to this problem:
When applying to jobs, use your preferred name. The top line of my resume is
Jean-Bernard Pellerin (JB)
I sign my emails as JB, and it's how I introduce myself. Upon meeting new people, sometimes they have seen my name written and are expecting me, so they'll say "hello Gene Burn-herd" which is a perfect time to interject and just say "I go by JB, it's easier for everyone". In your case, you can simply politely mention that you go by Bill and nobody will mind.
You'll find many employers already have systems in place for "nicknames". They deal with employees changing their names for marriage or whatever reason, so the ERP software can handle it quite easily. HR or payroll is also well-versed in aliases and will usually have forms you fill out that are separate from the other systems. So When I started at my current job, I applied as Jean-Bernard Pellerin, corresponded as JB, and finally filled out some paperwork as Jean-Bernard Landry-Pellerin. When I showed up on my first day, the chat clients were set up for me with JB, my email was some form of jPellerin, and my tax forms were filled out correctly. Like magic.
When applying somewhere smaller, I wouldn't automatically expect HR to have it's act together. In such a case, when accepting the job, I would reach out to the hiring manager and notify them of the challenges surrounding my name. They'll make sure it's handled differently with payroll and IT.
William Clinton goes by Bill, so for your case to be unprofessional would need significant divergence. I would avoid names which derive from inside-jokes, tribalism, counter-culture, or anything you wouldn't want associated with your professional life. When referring to tribalism I don't mean racial background, but avoid gang-names, sport nicknames, xXx_reefer_420_xXx, or going by maverick, goose, or ice-man.
Any derivation of your name, whether it be a shortening or a shift for language, should be just fine.
I hope this answer adds to the others and isn't just repetition, but I felt I could phrase it from experience and I hope it helps.