Hey there and welcome to the forums!
Your best bet would be to ask AMEX themselves about this sort of situation.
YES, AMEX can do what is called a Product Change and even without a hard pull on your credit and YES, it is possible to transfer the MR points along with it, BUT it is subject to their own discretion as well and everyone's situations are different so your best bet would be to call AMEX and express interest in a Product Change and ask them for the information you need - especially since you're going from a business card to a personal card.
That said, from my own experience, I was able to PC from a normal Blue card to an Everyday card and while they both use the Membership Rewards program, the two cards were on slightly different programs and I was still able to transfer my points on over.
AMEX: Everyday (MR), Macy's (cobranded)
MASTER: Citibank Dividend Platinum Select (non-World version)
VISA: Chase Amazon Signature, Chase (bank issued)
GE: Care Credit (medical expenses), Macy's (store), JCP (store)
AMEX: Costco True Earnings
VISA: Chase Ink Cash