The nature of this questions is shaky.

Do you want an entire strategy outlined here?

Are those 5K friends relevant at all? Are they targeted and engaged followers?
With organic reach so low, I wouldn't be excited if those 5K people went straight over to his fan page.

In the end, what does that mean?

(Ok, enough rhetorical questions )

My advice is to lay out a strategy that actually helps the client reach his/her goals rather than simply "change to a page."

I would create that page asap, outline a content strategy, and deploy an ad strategy to get super-specifically-targeted people into the pediatrician's funnel.
(Plus, it's always helps to show quantitative results to clients. Email addresses are almost as good as new patients. If you're super savvy, you can actually convert those new emails addresses into patients for him/her.)
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Start by getting enough likes to advertise.
Start advertising "shareable content" like helpful info in picture format. (People will share and engage with this)
If there are blog posts to promote, promote those and pixel your visitors.
If you can organize some sort of Facebook event or virtual giveaway, that will be a great way to generate quality leads for a pediatrician.
(Again, pixel people every chance you get so that you are able to re-target them and keep your marketing message in front of them.)

The most important thing here is to stay focused on delivering tangible results with social media services.

He needs to set up a business page so that he can begin advertising on facebook. You can't advertise from a personal page. Once he has the business page set up, he can use the feature on his business page "Build Audience" to "invite" all of the people on his personal page to "LIKE" his business page. It's a long process to convert all of those people over, but it can be done. He then needs to send an individual message to each of the people on his personal page letting them know he's moving to a business page. He also needs to start posting on his personal page that he's moving his business to a new business page and that he needs everyone to go to that page for all future facebook updates. Provide the link to the new business page. Just keep repeating the same message on the personal page for a couple of days and he should find that his personal friends will move over to being fans of his business page.

Create the new Page and then budget for a series of Promoted Posts to his Friends to go Like that page. Promoted Posts will get more reach. He's not going to be able to move them all over but he'll get a good chunk of them.