What incentive is there for a miner to not include a bad transaction?
Any block that contains an invalid transaction will be considered invalid as a whole, and every other peer/miner will reject it. A miner must check that every transaction included in a block is valid, otherwise there is a high risk that all the hard work required to solve the block is just being wasted.

Any miner who disobeys the validity requirements (by considering some invalid block as valid, or some valid block as invalid) will be completely disregarded by the rest of the network, defeating the point of any mining activity. 

Note: Sending 1BTC with a fee of 1BTC is a perfectly valid transaction (even though it might be a mistake by A). Of course, doing that, A must own 2BTC and give them away, or the transaction can't possibly be valid.