I don't see the need for your function F. People could just send to any agreed valid address which doesn't have a known public key associated with it such as 11111111111111111111BZbvjr. Spends to this address would generate magical crediting transaction on the "bettercoin" system.
A more difficult problem is managing the handover of mining power ie what to set the difficulty to on the new network and how to ramp it up as mining pools join. Also on the old network, how to prevent attacks when most miners have left but there are some normal users wanting to leave.
ByteCoin
The reason you need a function is because without it you'll need a database full of matching keys to figure out which bitcoin "sinkhole" address corresponds to the bettercoin address. This is unnecessary much overhead. Whereas a hash function can be computed in microseconds and has no storage requirements.
The weakness of the dying bitcoin network will be an issue which I haven't resolved. The maintainers of the bettercoin p2p network will have to decide how to treat the blockchain.
The exchange rate would just be 1:1 or perhaps 1:100 would be nice if bitcoin were @ .01 = $1. It doesn't really matter though as long as the rate is fixed.