AMEX will only allow a max of $5K on a BT so you will have to look elsewhere. I would look for a no-fee CU card like the NFCU Platinum to fill in the gaps but CUs are notoriously DTI-focused so your current debt load may be a bit limiting in the SL department.
Get a $15k loan from the CU.
Move the $10k off Discover to the CU.
Move $5k from the Chase cards to the CU.
BT the remaining $10k from the Chase cards to Discover.