Can I use a 150A cable to feed through a 200A breaker subpanel?
The conductors will have to be full size to the 200A breaker. Remember what protects the conductor is the 200A breaker if you reduce the conductor size there must be an additional breaker to protect the the current carrying conductor. NEC does allow for the feeder to be reduced for the final 1/3 distance of a feeder which is probably impractical, but it does allow the subpanel to have 150A buss.