question about copyrighting a version of a public domain book

Not so fast.

For the story, which you say is in the public domain, you don't need to make a fair use argument. For other versions, say, the Disney animated classic, you may need to, if your work is derivative of theirs. And fair use is generally not going to happen when your work is being produced for a commercial purpose.

Furthermore, your information on Peter Pan being in the public domain is incomplete - because parts of it definitely are not.
https://www.plagiarismtoday.com/2015/10/21/peter-pan-and-the-copyright-that-never-grew-up/
http://www.gosh.org/about-us/peter-pan/faqs

You'd best contact an attorney who specializes in copyright before you proceed.

Well, my goodness. Who knew?

I've been hip-deep in copyright law for the past year (with reference to Star Trek fan films). Fair use is not some magic guarantee people can hide behind, and public domain in the US is not the same as public domain elsewhere. Plus with the internet making pretty much every business an international one, content creators have to be aware of these things.

JM Barrie bequeathed the rights to Great Ormond's Street Hospital. Although pantomimes no longer have to give money to GOSH almost all of them do because it's seen as the right thing to do and not doing so would be a PR disaster.

Even Disney donates to GOSH. The copyright ended about the same time Disney partnered GOSH.

http://www.gosh.org/get-involved/corporate-fundraising/meet-our-supporters/walt-disney-company-ltd

Bad publicity could be worse than legal action.