Indeed , BA has no policy regarding offering protection on separate tickets

You may just have to purchase another ticket to LIS if IB will not move you free of charge. If you cannot make the connection, I would not think that the check in agent would be able to check bags through

You can ask at SFO if they can assist in getting to destination, but do not be surprised if you get referred to IB, What is the change policy on your IB ticket ? Can you change it now to a different fight