Go with whatever you want. Heck, it doesn't even have to be out of those two. You can try Fred, Jason, Gary, etc.

I still think you should ditch the William and just go with Liam for many reasons.  You are your own person, not a shadow of someone else.  Are they copacetic with this?  If they want it, then I'd use William just to please them and use Liam as your everyday name.

I like William too :3
But in the end, you're the one who's going to have to answer to it.
But don't worry about deciding right away.
I tried tons of names before I found the 'right' one :3
Until you get it legally changed, you can mess around with a whole bunch of them until one really feels right to you <3
Best of luck, dear and warm wishes!