I don't think it'll work. It's difficult to say with certainty because I'm not sure how the flasher will behave. If it's the flasher unit I'm familiar with, it requires 42 W of load for it to flash properly, it looks like you are only driving a relay with it.
I'm puzzled why you think the fog light ought to flash while reversing, rather than to be on solid. I often use fog light to supplement the reversing light, but that's because the reverse lights are not very bright on my car.