Yes, you can re-register your pool using the new stake keys associated with a new payment keypair (but be sure to register that new stake address first before re-registering your pool).
The challenge to consider is maintaining your pledge - to do this, I would register two entries for the owner pledge address - so it would look something like this in your pool re-registration:
–pool-owner-stake-verification-key-file old-stake.vkey \
Do not pay the 500 Ada fee when re-registering a pool, and make sure to use the same pool cold keys for signing.
Then you will need to wait for the new registration to take effect before moving your pledge from the old address to the new address.
I hope this helps.
Your friend, FROG